A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).
1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563
Offset: 0
Examples
E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! + (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
3, 6, 11, 18, 27, 38, ...
13, 26, 51, 94, 161, 258, ...
75, 150, 299, 582, 1083, 1910, ...
541, 1082, 2163, 4294, 8345, 15666, ...
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
- N. J. A. Sloane, Transforms
Crossrefs
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 50); T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >; A292915:= func< n,k | T(k,n-k) >; [A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024 -
Maple
A:= proc(n, k) option remember; k^n +add( binomial(n, j)*A(j, k), j=0..n-1) end: seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 27 2017 -
Mathematica
Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten -
PARI
a000670(n) = sum(k=0, n, k!*stirling(n, k, 2)); A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023
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SageMath
def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n] def A292915(n,k): return T(k,n-k) flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024
Formula
E.g.f. of column k: exp(k*x)/(2 - exp(x)).
A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - Seiichi Manyama, Dec 25 2023
Comments