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A174264 A triangle of coefficients on infinite sum polynomials: p(x,n)=If[n == 0, 1, (1 - x)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^ k, {k, 0, Infinity}]/x] 1
1, 1, 1, 1, 18, 42, 18, 1, 1, 115, 1539, 5065, 5065, 1539, 115, 1, 1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1, 1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 80, 13440, 5913600, 5381376000, 8782405632000, 23361198981120000,

94566133475573760000, 553211880832106496000000, 4492080472356704747520000000,...}.

LINKS

Table of n, a(n) for n=0..37.

FORMULA

p(x,n)=If[n == 0, 1, (1 - x)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^ k, {k, 0, Infinity}]/x];

out_n,m=coefficients(p(x,n))

EXAMPLE

{1},

{1, 1},

{1, 18, 42, 18, 1},

{1, 115, 1539, 5065, 5065, 1539, 115, 1},

{1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1},

{1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256, 487944, 3109, 1},

{1, 15606, 7232832, 588609722, 15102054532, 159360510654, 796011579264, 2034786608786, 2770692409206, 2034786608786, 796011579264, 159360510654, 15102054532, 588609722, 7232832, 15606, 1},

{1, 78103, 103694985, 19568948247, 1065525448614, 23072731441362, 236032579067166, 1262043871882890, 3749020958436984, 6409344151561648, 6409344151561648, 3749020958436984, 1262043871882890, 236032579067166, 23072731441362, 1065525448614, 19568948247, 103694985, 78103, 1},

{1, 390600, 1466023731, 619322458800, 67773276182575, 2802617455410216, 53645573041228725, 536366226569480256, 3023314553367761850, 10090695137544912400, 20563892762682272046, 26024562946121517600, 20563892762682272046, 10090695137544912400, 3023314553367761850, 536366226569480256, 53645573041228725, 2802617455410216, 67773276182575, 619322458800, 1466023731, 390600, 1},

{1, 1953097, 20606359662, 19105228968022, 4062046061251702, 306352064179097622, 10355782284092172382, 180449348295691590742, 1772697944064120724647, 10422061778244020252047, 38179816523099760064252, 89512894147925375525772, 136527354458904110040052, 136527354458904110040052, 89512894147925375525772, 38179816523099760064252, 10422061778244020252047, 1772697944064120724647, 180449348295691590742, 10355782284092172382, 306352064179097622, 4062046061251702, 19105228968022, 20606359662, 1953097, 1},

{1, 9765594, 288951921066, 581527646706874, 235124431637251555, 31362739743718489620, 1800493681167699729940, 52143903931162149522580, 843767280698373213383505, 8174015411024678270125030, 49739731928304174504355510, 196644863528988544778420550, 517044213844787972256968395, 918147140681329965091033240, 1110785055014320687664653080, 918147140681329965091033240, 517044213844787972256968395, 196644863528988544778420550, 49739731928304174504355510, 8174015411024678270125030, 843767280698373213383505, 52143903931162149522580, 1800493681167699729940, 31362739743718489620, 235124431637251555, 581527646706874, 288951921066, 9765594, 1}

MATHEMATICA

p[x_, n_] = If[n == 0, 1, (1 - x)^(3*n + 1)*Sum[(k*( k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%]

CROSSREFS

Cf. A060187, A154283

Sequence in context: A285527 A097972 A154284 * A124792 A090605 A318168

Adjacent sequences:  A174261 A174262 A174263 * A174265 A174266 A174267

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 14 2010

STATUS

approved

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Last modified January 19 15:37 EST 2020. Contains 331049 sequences. (Running on oeis4.)