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A156136
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A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x).
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0
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1, 2, 2, 3, 12, 4, 4, 36, 48, 8, 5, 80, 240, 160, 16, 6, 150, 800, 1200, 480, 32, 7, 252, 2100, 5600, 5040, 1344, 64, 8, 392, 4704, 19600, 31360, 18816, 3584, 128, 9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256, 10, 810, 17280, 141120, 508032
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OFFSET
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0,2
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COMMENTS
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Row sums are:
{1, 4, 19, 96, 501, 2668, 14407, 78592, 432073, 2390004,...}.
I tried this sum because it is a double binomial
like the Narayana numbers of the first kind.
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 75-76
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LINKS
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Table of n, a(n) for n=0..49.
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FORMULA
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p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
p(x,n)=n Hypergeometric2F1[1 - n, 1 - n, 2, 2 x];
t(n,m)=coefficiemts(p(x,n))
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EXAMPLE
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{1},
{2, 2},
{3, 12, 4},
{4, 36, 48, 8},
{5, 80, 240, 160, 16},
{6, 150, 800, 1200, 480, 32},
{7, 252, 2100, 5600, 5040, 1344, 64},
{8, 392, 4704, 19600, 31360, 18816, 3584, 128},
{9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256},
{10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512}
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MATHEMATICA
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Clear[t0, p, x, n, m];
p[x_, n_] = Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]
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CROSSREFS
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A142983, A142978
Sequence in context: A143933 A075095 A178343 * A134243 A182779 A199673
Adjacent sequences: A156133 A156134 A156135 * A156137 A156138 A156139
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KEYWORD
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nonn,tabl,uned
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AUTHOR
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Roger L. Bagula, Feb 04 2009
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STATUS
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approved
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