The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A157521 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}]. 0
 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; text; internal format)
 OFFSET 0,4 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. LINKS 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}]. 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} 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[%] CROSSREFS Sequence in context: A191619 A327245 A157525 * A176005 A211963 A275080 Adjacent sequences:  A157518 A157519 A157520 * A157522 A157523 A157524 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula, Mar 02 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 03:33 EST 2021. Contains 349562 sequences. (Running on oeis4.)