Search: a006304 -id:a006304
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A006306
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Coefficients of the '2nd-order' mock theta function mu(q).
(Formerly M0163)
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+10
4
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1, -1, 1, 2, -1, -4, 1, 5, -2, -5, 4, 7, -4, -11, 3, 13, -6, -14, 9, 18, -7, -24, 8, 29, -14, -32, 17, 38, -18, -50, 20, 58, -25, -63, 33, 77, -35, -94, 36, 108, -48, -122, 60, 141, -63, -170, 70, 195, -87, -215, 101, 250, -110, -294, 124, 333, -146, -371, 173, 424, -190, -492, 206, 554, -245, -617, 283
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OFFSET
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0,4
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COMMENTS
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Coefficients of the "second-order" mock theta function mu(q).
|a(n)| is the number of partitions of n without repeated odd parts whose M2-rank is even minus the number of partitions of n without repeated odd parts whose M2-rank is odd. (End)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: Sum_{n >= 0} (-1)^n q^n^2 (1-q)(1-q^3)...(1-q^(2n-1))/((1+q^2)^2 (1+q^4)^2 ... (1+q^(2n))^2).
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EXAMPLE
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G.f. = 1 - x + x^2 + 2*x^3 - x^4 - 4*x^5 + x^6 + x*x^7 - 2*x^8 - 5*x^9 + ...
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MATHEMATICA
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CoefficientList[Series[Sum[(-q)^n^2 Product[(1-q^(2k-1))/(1+q^(2k))^2, {k, 1, n}], {n, 0, 10}], {q, 0, 100}], q]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[- x^2, x^2, k]^2, {k, 0, Sqrt[ n]}], {x, 0, n}]]; (* Michael Somos, Jul 09 2015 *)
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A006305
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Taylor series related to one in Ramanujan's Lost Notebook.
(Formerly M1014)
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+10
3
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1, 2, 4, 6, 10, 16, 25, 38, 58, 84, 122, 174, 244, 338, 465, 630, 850, 1136, 1508, 1988, 2608, 3398, 4408, 5688, 7306, 9342, 11900, 15090, 19070, 24008, 30122, 37666, 46955, 58348, 72302, 89338, 110094, 135316, 165912, 202924, 247632, 301508
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} q^(n^2+n) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)^2 (1-q^2) (1-q^3)^2 (1-q^4) ... (1-q^(2n)) (1-q^(2n+1))^2).
a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2.74858241446108527... and c = 0.1051685561271293027... - Vaclav Kotesovec, Jun 12 2019
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EXAMPLE
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G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 58*x^8 + ...
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MATHEMATICA
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Series[Sum[q^(n^2+n)/(1-q)^2 Product[(1+q^(2k))/((1-q^(2k))(1-q^(2k+1))^2), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k k + k) QPochhammer[ -x^2, x^2, k] / (QPochhammer[ x, x, 2 k + 1] QPochhammer[ x, x^2, k + 1] ) , {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jul 09 2015 *)
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/(1-x)^2 * Product[(1+x^(2*j))/((1-x^(2*j))*(1-x^(2*j+1))^2), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A153140
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Coefficients of the second order mock theta function B(q).
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+10
1
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1, 2, 4, 6, 9, 14, 20, 28, 40, 54, 72, 98, 129, 168, 220, 282, 360, 460, 580, 728, 912, 1134, 1404, 1734, 2129, 2604, 3180, 3864, 4680, 5658, 6812, 8182, 9808, 11718, 13968, 16618, 19720, 23350, 27600, 32550, 38313
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{n >= 0} q^(n^2+n)(1+q^2)(1+q^4)...(1+q^(2n))/(1-q)^2(1-q^3)^2...(1-q^(2n+1))^2.
G.f.: Sum_{n >= 0} q^n(1+q)(1+q^3)...(1+q^(2n-1))/(1-q)(1-q^3)...(1-q^(2n+1)).
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) * Product[(1+x^(2*j))/(1-x^(2*j+1))^2, {j, 0, k}], {k, 0, Floor[Sqrt[nmax]]}]/2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
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PROG
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(PARI) lista(nn) = my(q = qq + O(qq^nn)); gf = sum(n = 0, nn, q^n * prod(k = 1, n, 1 + q^(2*k-1)) / prod(k = 0, n, 1 - q^(2*k+1))); Vec(gf) \\ Michel Marcus, Jun 18 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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