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A292913 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(k*x)-1). 1
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 15, 0, 1, 5, 32, 135, 240, 52, 0, 1, 6, 50, 320, 1215, 1664, 203, 0, 1, 7, 72, 625, 3840, 12636, 12992, 877, 0, 1, 8, 98, 1080, 9375, 53248, 147987, 112256, 4140, 0, 1, 9, 128, 1715, 19440, 162500, 831488, 1917999, 1059840, 21147, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

FORMULA

O.g.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - k*x/(1 - 3*k*x/(1 - k*x/(1 - 4*k*x/(1 - ...))))))))), a continued fraction.

E.g.f. of column k: exp(exp(k*x)-1).

A(n,k) = exp(-1)*k^n*Sum_{j>=0} j^n/j!.

A(n,k) = k^n * Bell(n). - Alois P. Heinz, Sep 26 2017

EXAMPLE

E.g.f. of column k: A_k(x) =  1 + k*x/1! + 2*k^2*x^2/2! + 5*k^3*x^3/3! + 15*k^4 x^4/4! + 52*k^5*x^5/5! + ...

Square array begins:

1,   1,     1,      1,      1,       1,  ...

0,   1,     2,      3,      4,       5,  ...

0,   2,     8,     18,     32,      50,  ...

0,   5,    40,    135,    320,     625,  ...

0,  15,   240,   1215,   3840,    9375,  ...

0,  52,  1664,  12636,  53248,  162500,  ...

MAPLE

A:= (n, k)-> k^n * combinat[bell](n):

seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 26 2017

MATHEMATICA

Table[Function[k, n! SeriesCoefficient[Exp[Exp[k x] - 1], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-((-1)^(i + 1) (i - 1) + i + 3) k x/4, 1, {i, 0, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

CROSSREFS

Columns k=0..3 give A000007, A000110, A055882, A247452.

Rows n=0..2 give A000012, A001477, A001105.

Main diagonal gives A292914.

Sequence in context: A306024 A237018 A290605 * A214776 A317575 A295653

Adjacent sequences:  A292910 A292911 A292912 * A292914 A292915 A292916

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Sep 26 2017

STATUS

approved

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Last modified July 27 14:35 EDT 2021. Contains 346306 sequences. (Running on oeis4.)