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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A(n,k) is the k-th binomial transform of A000670 evaluated at n.

LINKS

Table of n, a(n) for n=0..65.

N. J. A. Sloane, Transforms

FORMULA

E.g.f. of column k: exp(k*x)/(2 - exp(x)).

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

CROSSREFS

Columns k=0..3 give A000670, A000629, A007047, A259533.

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

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Last modified August 3 06:43 EDT 2021. Contains 346435 sequences. (Running on oeis4.)