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

0,5

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

Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.

Rows n=0..2 give A000012, A000027, A102305.

Main diagonal gives A292916.

Sequence in context: A059481 A113592 A271702 * A271700 A136555 A343627

Adjacent sequences: A292912 A292913 A292914 * A292916 A292917 A292918

Keyword

nonn,tabl

Author

_Ilya Gutkovskiy_, Sep 26 2017

Status

approved