OFFSET
0,5
COMMENTS
A(n,k) is the k-th binomial transform of A000670 evaluated at n.
LINKS
G. C. Greubel, Antidiagonals n = 0..50, flattened
N. J. A. Sloane, Transforms
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
EXAMPLE
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, ...
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
PROG
(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
(Magma)
R<x>:=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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Sep 26 2017
STATUS
approved