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A178252
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Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the umbral inverse Bernoulli polynomials B^{-1}(n,x), 0 <= m <= n.
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4
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1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 5, 10, 5, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 9, 12, 21, 126, 21, 12, 9, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 11, 55, 165, 66, 77, 66, 165, 55, 11, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1
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OFFSET
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0,9
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COMMENTS
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The fractions A053382(n,m)/A053383(n,m) give the triangle of the coefficients of the Bernoulli polynomials:
1;
-1/2, 1;
1/6, -1, 1;
0, 1/2, -3/2, 1;
-1/30, 0, 1, -2, 1;
0, -1/6, 0, 5/3, -5/2, 1;
1/42, 0, -1/2, 0, 5/2, -3, 1;
The matrix inverse of this triangle defines coefficients of the umbral inverse Bernoulli polynomials B^{-1}(n,x) in row n:
1;
1/2, 1;
1/3, 1, 1;
1/4, 1, 3/2, 1;
1/5, 1, 2, 2, 1;
1/6, 1, 5/2, 10/3, 5/2, 1;
1/7, 1, 3, 5, 5, 3, 1;
1/8, 1, 7/2, 7, 35/4, 7, 7/2, 1;
1/9, 1, 4, 28/3, 14, 14, 28/3, 4, 1;
1/10, 1, 9/2, 12, 21, 126/5, 21, 12, 9/2, 1;
1/11, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1;
The current triangle T(n,m) is the numerator of the entry in row n and column m.
In the majority of cases, T(n,m) = A050169(n,m), but since we use the numerators of the reduced fractions, an integer factor may be missing in this equation.
Umbral composition (e.g., B(.,x)^k = B(k,x)) gives B^(-1)(n,B(.,x)) = x^n = B(n,B^(-1)(.,x)). - Tom Copeland, Aug 25 2015
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LINKS
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FORMULA
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"Palindromic:" T(n,m+1) = T(n,n-m). T(n,0)=1.
The umbral inverse Bernoulli polynomials are Binv(n,x) = [(1+x)^(n+1)-x^(n+1)]/(n+1) with the e.g.f. e^(t*x) * (e^t-1)/t. (See A074909 for more details.) Therefore, T(n,k) is the numerator of the reduced fraction C(n+1,k)/(n+1) for 0 <= k < (n+1).
The reversed rows are presented as the diagonals of A258820.
T(n,k) = A258820(2n-k,n-k) = A003989(n+1,n+1-k) * n! / [ k! (n+1-k)! ], where A003989(j,k) = gcd(j,k). (End)
The following refers to the rational triangle TBinv with entries T(n,k)/A178340(n, m), n >= m >= 0.
The inverse of the Bernoulli triangle TB(n, m) with entries A196838(n,m)/A196839(n,m), n >= m >= 0, is the Sheffer triangle (z/(exp(z)-1),z). Therefore, the inverse triangle TBinv is the Sheffer triangle ((exp(z)-1)/z, z). This means that the e.g.f. of the sequence of column m of TBinv ((exp(x)-1)/x)*x^m/m! for m >= 0.
The e.g.f. of the row polynomials of TBinv, called Binv(n, x) = Sum_{m=0..n} TBinv(n,m)*x^m, is gBinv(z,x) = ((exp(z)-1)/z)*exp(x*z) (of the so-called Appell type).
The e.g.f. of the row sums is gBinv(x,1).
The e.g.f. of the alternating row sums is gBinv(x,-1) = (1 - exp(-x))/x.
The e.g.f. of the a-sequence of this Sheffer triangle is 1, and the e.g.f. of the z-sequence is (exp(x) - x -1)/((exp(x) -1)*x). This is the sequence 1/2, -1/12, 0, 1/120, 0, -1/252, 0, 1/240, 0, -1/132, .... For a- and z-sequences of Sheffer triangles and the corresponding recurrences see A006232.
The convolution property of the row polynomials Binv(n, x) is Binv(n, x+y) = Sum_{k=0..n} binomial(n, k)*Binv(n-k, x)*y^n (or with x and y exchanged).
The row polynomials satisfy (d/dx)Binv(n, x) = n*Binv(n-1, x), with Binv(0, x) = 1 (from Meixner's identity).
(End)
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EXAMPLE
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The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13
0: 1
1: 1 1
2: 1 1 1
3: 1 1 3 1
4: 1 1 2 2 1
5: 1 1 5 10 5 1
6: 1 1 3 5 5 3 1
7: 1 1 7 7 35 7 7 1
8: 1 1 4 28 14 14 28 4 1
9: 1 1 9 12 21 126 21 12 9 1
10: 1 1 5 15 30 42 42 30 15 5 1
11: 1 1 11 55 165 66 77 66 165 55 11 1
12: 1 1 6 22 55 99 132 132 99 55 22 6 1
13: 1 1 13 26 143 143 429 1716 429 143 143 26 13 1
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MAPLE
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nm := 15 : eM := Matrix(nm, nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1, m+1] :=coeff(bernoulli(n, x), x, m) ; end do: for m from n+1 to nm-1 do eM[n+1, m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%a, ", numer(eM[n, m])) ; end do: end do: # R. J. Mathar, Dec 21 2010
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MATHEMATICA
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max = 13; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes]; Table[ Take[inv[[n]], n], {n, 1, max}] // Flatten // Numerator (* Jean-François Alcover, Aug 09 2012 *)
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PROG
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(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(numerator(binomial(n+1, k)/(n+1)), ", "); ); print(); ); } \\ after Tom Copeland comment; Michel Marcus, Jul 25 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Redefined based on reduced fractions by R. J. Mathar, Dec 21 2010
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STATUS
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approved
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