A334192 - OEIS

A334192

Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (k*j + 1)^n / ((-k)^j * j!).

1, 1, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180

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OFFSET

0,9

LINKS

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

FORMULA

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (-x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).

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

EXAMPLE

Square array begins:

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

0, 0, 0, 0, 0, 0, ...

-1, -2, -3, -4, -5, -6, ...

-1, -4, -9, -16, -25, -36, ...

2, 4, 0, -16, -50, -108, ...

9, 64, 189, 384, 625, 864, ...

MATHEMATICA

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

Table[Function[k, n! SeriesCoefficient[Exp[x + (1 - Exp[k x])/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

CROSSREFS

Columns k=1..3 give A293037, A334190, A334191.

Cf. A309386, A334165, A334193 (diagonal).

Sequence in context: A238349 A318754 A318758 * A124790 A325734 A351322

Adjacent sequences: A334189 A334190 A334191 * A334193 A334194 A334195

KEYWORD

sign,tabl

AUTHOR

Ilya Gutkovskiy, Apr 18 2020

STATUS

approved