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

 

Logo

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

Table of n, a(n) for n=0..42.

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.

License Agreements, Terms of Use, Privacy Policy. .

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