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A156918
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An infinite sum polynomial coefficients triangle: p(x,n)=((1 + x - x^2)^ (n + 1))*Sum[(2*k + 1)^n*(-x + x^2)^k, {k, 0, Infinity}].
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0
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1, 1, -1, 1, 1, -6, 7, -2, 1, 1, -23, 46, -47, 26, -3, 1, 1, -76, 306, -536, 459, -232, 82, -4, 1, 1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1, 1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1, 1, -2179, 62836
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row sums are one.
This polynomials set is MacMahon level in Fibonacci type polynomials.
Second column is MacMahon like:
{6, 23, 76, 237, 722, 2179, 6552, 19673, 59038,...}.
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FORMULA
| p(x,n)=((1 + x - x^2)^ (n + 1))*Sum[(2*k + 1)^n*(-x + x^2)^k, {k, 0, Infinity}];
t(n,m)=coefficients(p(x,n)).
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EXAMPLE
| {1},
{1, -1, 1},
{1, -6, 7, -2, 1},
{1, -23, 46, -47, 26, -3, 1},
{1, -76, 306, -536, 459, -232, 82, -4, 1},
{1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1},
{1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1},
{1, -2179, 62836, -381037, 1099549, -1878718, 2123525, -1658537, 898985, -346886, 93377, -13109, 2200, -7, 1},
{1, -6552, 338164, -3148512, 12462490, -28641208, 43293424, -45549160, 34547939, -19196280, 7688816, -2219048, 469274, -45920, 6580, -8, 1},
{1, -19673, 1776013, -25220652, 136293550, -412190386, 806962178, -1103790288, 1101479867, -818266435, 457055627, -192821456, 59968306, -13396194, 2307310, -157468, 19709, -9, 1},
{1, -59038, 9175179, -196533186, 1450942135, -5689688308, 14162226218, -24518458404, 31038919661, -29591235130, 21627256629, -12194700606, 5294331373, -1766049132, 440991850, -77888572, 11241719, -531462, 59083, -10, 1}
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MATHEMATICA
| Clear[p, x, n, m];
p[x_, n_] = ((1 + x - x^2)^ (n + 1))*Sum[(2*k + 1)^n*(-x + x^2)^k, {k, 0, Infinity}];
Table[Expand[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A011483 A201323 A155490 * A123154 A021602 A198098
Adjacent sequences: A156915 A156916 A156917 * A156919 A156920 A156921
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 18 2009
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