A326479 - OEIS

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A326479

T(n, k) = 2^n * n! * [x^k] [z^n] (exp(z) + 1)^2/(4*exp(x*z)), triangle read by rows, for 0 <= k <= n.

1, 2, -2, 6, -8, 4, 20, -36, 24, -8, 72, -160, 144, -64, 16, 272, -720, 800, -480, 160, -32, 1056, -3264, 4320, -3200, 1440, -384, 64, 4160, -14784, 22848, -20160, 11200, -4032, 896, -128, 16512, -66560, 118272, -121856, 80640, -35840, 10752, -2048, 256

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..44.

FORMULA

Generated by 1/A326480(z), where A326480(z) denotes the generating function of A326480 which generates the Euler polynomials of order 2.

EXAMPLE

[0] [ 1]

[1] [ 2, -2]

[2] [ 6, -8, 4]

[3] [ 20, -36, 24, -8]

[4] [ 72, -160, 144, -64, 16]

[5] [ 272, -720, 800, -480, 160, -32]

[6] [ 1056, -3264, 4320, -3200, 1440, -384, 64]

[7] [ 4160, -14784, 22848, -20160, 11200, -4032, 896, -128]

[8] [16512, -66560, 118272, -121856, 80640, -35840, 10752, -2048, 256]

[9] [65792, -297216, 599040, -709632, 548352, -290304, 107520, -27648, 4608, -512]

MAPLE

IE2 := proc(n) (exp(z) + 1)^2/(4*exp(x*z));

series(%, z, 48); 2^n*n!*coeff(%, z, n) end:

for n from 0 to 9 do PolynomialTools:-CoefficientList(IE2(n), x) od;

MATHEMATICA

T[n_, k_] := 2^n n! SeriesCoefficient[(E^z + 1)^2/(4 E^(x z)), {x, 0, k}, {z, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

CROSSREFS

Cf. A326480, A063376.

Sequence in context: A101343 A284748 A134457 * A306688 A092522 A116542

Adjacent sequences: A326476 A326477 A326478 * A326480 A326481 A326482

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jul 12 2019

STATUS

approved