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A124320
Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.
7
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
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
Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).
Sequence in context: A122761 A100469 A360857 * A156146 A192043 A154584
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 26 2006
STATUS
approved