A124320 Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.

1, 1, 1, 1, 2, 6, 1, 3, 12, 60, 1, 4, 20, 120, 840, 1, 5, 30, 210, 1680, 15120, 1, 6, 42, 336, 3024, 30240, 332640, 1, 7, 56, 504, 5040, 55440, 665280, 8648640, 1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800

Offset

0,5

Comments

This is the Pochhammer function which is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1. Also known as the rising factorial power. - Peter Luschny, Jan 09 2011

References

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol

Formula

T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - Peter Luschny, Jan 09 2011

Example

Triangle starts:
[0]  1;
[1]  1, 1;
[2]  1, 2,   6;
[3]  1, 3,  12,  60;
[4]  1, 4,  20, 120,  840;
[5]  1, 5,  30, 210, 1680, 15120;
[6]  1, 6,  42, 336, 3024, 30240, 332640;
[7]  1, 7,  56, 504, 5040, 55440, 665280, 8648640;
Array starts:
[0] 1,  1,   6,   60,   840,   15120,   332640,   8648640, ... A000407
[1] 1,  2,  12,  120,  1680,   30240,   665280,  17297280, ... A001813
[2] 1,  3,  20,  210,  3024,   55440,  1235520,  32432400, ... A006963
[3] 1,  4,  30,  336,  5040,   95040,  2162160,  57657600, ... A001761
[4] 1,  5,  42,  504,  7920,  154440,  3603600,  98017920, ... A102693
[5] 1,  6,  56,  720, 11880,  240240,  5765760, 160392960, ... A093197
[6] 1,  7,  72,  990, 17160,  360360,  8910720, 253955520, ... A203473
[7] 1,  8,  90, 1320, 24024,  524160, 13366080, 390700800, ...
[8] 1,  9, 110, 1716, 32760,  742560, 19535040, 586051200, ...
[9] 1, 10, 132, 2184, 43680, 1028160, 27907200, 859541760, ...

Maple

T:=proc(n, k) if k<=n then binomial(n+k-1, k)*k! else 0 fi end: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
A124320 := (n, k)-> `if`(n=0 and k=0, 1, pochhammer(n, k)); seq(print(seq(A124320(n, k), k=0..n)), n=0..5); # Peter Luschny, Jan 09 2011

Mathematica

Table[Pochhammer[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* Peter Luschny, Jan 09 2011 *)

Prog

(Sage)
for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # Peter Luschny, Jan 09 2011
(PARI) for(n=0, 10, for(k=0, n, print1(if(n==0 && k==0, 1, (n+k-1)!/(n-1)!), ", "))) \\ G. C. Greubel, Nov 19 2017

Crossrefs

Rows of array: A000407, A001813, A006963, A001761, A102693, A093197, A203473.

Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).

Sequence in context: A122761 A100469 A360857 * A156146 A192043 A154584

Adjacent sequences: A124317 A124318 A124319 * A124321 A124322 A124323

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 26 2006

Status

approved