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A157521
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A triangular function using Gamma functiona as Binomial like: t(n,m)=n!*d/dm*(Binomial[n,m]); g(x,y)=y!*D[Gamma[y + 1]/(Gamma[x + 1]*Gamma[y - x + 1]),{1,x}].
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0
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0, 1, -1, 3, 0, -3, 11, 9, -9, -11, 50, 80, 0, -80, -50, 274, 650, 400, -400, -650, -274, 1764, 5544, 6300, 0, -6300, -5544, -1764, 13068, 51156, 82908, 44100, -44100, -82908, -51156, -13068, 109584, 513792, 1072512, 1016064, 0, -1016064, -1072512
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row sums are zero since the derivative positive on one side and negative on the other.
I call this the "collapsing middle" binomial function.
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FORMULA
| t(n,m)=n!*d/dm*(Binomial[n,m]);
g(x,y)=y!*D[Gamma[y + 1]/(Gamma[x + 1]*Gamma[y - x + 1]),{1,x}].
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EXAMPLE
| {0},
{1, -1},
{3, 0, -3},
{11, 9, -9, -11},
{50, 80, 0, -80, -50},
{274, 650, 400, -400, -650, -274},
{1764, 5544, 6300, 0, -6300, -5544, -1764},
{13068, 51156, 82908, 44100, -44100, -82908, -51156, -13068},
{109584, 513792, 1072512, 1016064, 0, -1016064, -1072512, -513792, -109584},
{1026576, 5610384, 14276736, 18797184, 9144576, -9144576, -18797184, -14276736, -5610384, -1026576},
{10628640, 66369600, 198871200, 330739200, 279417600, 0, -279417600, -330739200, -198871200, -66369600, -10628640}
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MATHEMATICA
| f[x_, y_] = Gamma[y + 1]/(Gamma[x + 1]*Gamma[y - x + 1]);
g[x_, y_] = D[f[x, y], {x, 1}];
Table[Table[Rationalize[y!*N[g[x, y]]], {x, 0, y}], {y, 0, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A177785 A191619 A157525 * A176005 A128252 A033596
Adjacent sequences: A157518 A157519 A157520 * A157522 A157523 A157524
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 02 2009
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