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A171695 Expansion generating function using an infinite sum:m=1; f(t,y)=Sum[2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[ -2^m*t])*y^x, {x, 0, Infinity}] 0
1, 1, 1, -1, 6, -1, -1, 7, 25, -7, 10, -44, 152, -20, -2, -26, 198, -292, 1628, -642, 94, -154, 1000, -1954, 6416, 1586, -1400, 266, 1646, -13606, 51774, -75094, 175226, -73890, 15962, -1378, 1000, -3936, -4448, 190432, 37104, 779104, -472160, 133152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 4, 24, 96, 960, 5760, 80640, 645120, 11612160, 116121600,...}.

m=-1 gives MacMahon {1,6,1} A060187.

Higher and lower m values seem to diverge for symmetrical triangles.

LINKS

Table of n, a(n) for n=0..43.

FORMULA

Infinite sum on an generalized Euler numbers/ polynomial scaled generating function:

f(t,y)=Sum[2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[ -2^m*t])*y^x, {x, 0, Infinity}]

Scaling function is:

g(y,n)=(1 - y)^(n + 1)*2^(Floor[(n + 1)/2])*n!

EXAMPLE

{1},

{1, 1},

{-1, 6, -1},

{-1, 7, 25, -7},

{10, -44, 152, -20, -2},

{-26, 198, -292, 1628, -642, 94},

{-154, 1000, -1954, 6416, 1586, -1400, 266},

{1646, -13606, 51774, -75094, 175226, -73890, 15962, -1378},

{1000, -3936, -4448, 190432, 37104, 779104, -472160, 133152, -15128},

{-92744, 915368, -4000000, 10902272, -12924496, 23857424, -8350528, 1256000, 85208, -36344},

{302984, -3612496, 19683208, -62131328, 158998096, -155665568, 272115856, -157902848, 53634088, -10139536, 839144}

MATHEMATICA

Clear[m, n, t, x, y, a]

m = 1;

f[t_, y_] = Sum[2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[ -2^m* t])*y^x, {x, 0, Infinity}]

a = Table[ CoefficientList[FullSimplify[ExpandAll[(1 - y)^(n + 1)*2^(Floor[(n + 1)/2])*n!*SeriesCoefficient[ Series[f[t, y], {t, 0, 30}], n]]], y], {n, 0, 10}]

Flatten[a]

CROSSREFS

A159041, A060187, A171692, A171693, A171694

Sequence in context: A195397 A173741 A171147 * A179233 A141600 A303489

Adjacent sequences:  A171692 A171693 A171694 * A171696 A171697 A171698

KEYWORD

sign,uned

AUTHOR

Roger L. Bagula, Dec 15 2009

STATUS

approved

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Last modified March 30 09:49 EDT 2020. Contains 333125 sequences. (Running on oeis4.)