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

Author

Emeric Deutsch, Oct 26 2006

Keywords

• nonn
• tabl

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

Examples

			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, ...
		

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

Crossrefs

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

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

Programs

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 *)

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

Sage

for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # Peter Luschny, Jan 09 2011

Formula

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