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A154646
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A triangular sequence of coefficients: p(x,n)=(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}].
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1
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2, 3, 3, 5, 26, 5, 9, 153, 153, 9, 17, 796, 2262, 796, 17, 33, 3951, 25176, 25176, 3951, 33, 65, 19266, 243111, 524876, 243111, 19266, 65, 129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129, 257, 453848, 18445820, 127880936, 235517318
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Row sums are:
{2, 6, 36, 324, 3888, 58320, 1049760, 22044960, 529079040, 14285134080,
428554022400,...}.
This results from a modular form bilinear approach summed:
f1(x)=(3*x+1)/(-x); f2(x)=(3*x+2)/(-x).
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 27 2009: I got the previous result from trying to find an infinite sum for higher Sierpinski-Pascal levels: this version was found using the umbral expansion approach.
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FORMULA
| p(x,n)=(-1)^(-1 + n)* 3^n* (-1 + x)(1 - n) LerchPhi[x, -n, 1/3]+
(-1)^(-1 + n)* 3^n* (-1 + x)(1 - n) LerchPhi[x, -n, 2/3];
p(x,n)=(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}] +
(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];
t(n,m)=coefficients(p(x,n))
p(t,x)=Exp[t]*x/((-Exp[3*t] + x)) + Exp[2*t]*x/((-Exp[3*t] + x)) [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 27 2009]
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EXAMPLE
| {2},
{3, 3},
{5, 26, 5},
{9, 153, 153, 9},
{17, 796, 2262, 796, 17},
{33, 3951, 25176, 25176, 3951, 33},
{65, 19266, 243111, 524876, 243111, 19266, 65},
{129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129},
{257, 453848, 18445820, 127880936, 235517318, 127880936, 18445820, 453848, 257}, {513, 2210139, 152441730, 1711859886, 5276054772, 5276054772, 1711859886,152441730, 2210139, 513},
{1025, 10802926, 1237317237, 21613648728, 104609410314, 173611661940,104609410314, 21613648728, 1237317237, 10802926, 1025}
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MATHEMATICA
| Clear[p]; p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}]
+ (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 27 2009: (Start)
p[t_] = Exp[t]*x/((-Exp[3*t] + x)) + Exp[2*t]*x/((-Exp[3*t] + x));
a = Table[ CoefficientList[FullSimplify[ExpandAll[(n!*(-1 + x)^(n + 1)/x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];
Flatten[a] (End)
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CROSSREFS
| Sequence in context: A064776 A096659 A154695 * A046826 A054892 A104570
Adjacent sequences: A154643 A154644 A154645 * A154647 A154648 A154649
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KEYWORD
| nonn,uned,tabl
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 13 2009
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