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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.

%I A124320 #26 Mar 22 2022 09:52:55
%S A124320 1,1,1,1,2,6,1,3,12,60,1,4,20,120,840,1,5,30,210,1680,15120,1,6,42,
%T A124320 336,3024,30240,332640,1,7,56,504,5040,55440,665280,8648640,1,8,72,
%U A124320 720,7920,95040,1235520,17297280,259459200,1,9,90,990,11880,154440,2162160,32432400,518918400,8821612800
%N A124320 Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.
%C A124320 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
%D A124320 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
%H A124320 G. C. Greubel, <a href="/A124320/b124320.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A124320 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/5.2#iii">Pochhammer's Symbol</a>
%F A124320 T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - _Peter Luschny_, Jan 09 2011
%e A124320 Triangle starts:
%e A124320 [0]  1;
%e A124320 [1]  1, 1;
%e A124320 [2]  1, 2,   6;
%e A124320 [3]  1, 3,  12,  60;
%e A124320 [4]  1, 4,  20, 120,  840;
%e A124320 [5]  1, 5,  30, 210, 1680, 15120;
%e A124320 [6]  1, 6,  42, 336, 3024, 30240, 332640;
%e A124320 [7]  1, 7,  56, 504, 5040, 55440, 665280, 8648640;
%e A124320 Array starts:
%e A124320 [0] 1,  1,   6,   60,   840,   15120,   332640,   8648640, ... A000407
%e A124320 [1] 1,  2,  12,  120,  1680,   30240,   665280,  17297280, ... A001813
%e A124320 [2] 1,  3,  20,  210,  3024,   55440,  1235520,  32432400, ... A006963
%e A124320 [3] 1,  4,  30,  336,  5040,   95040,  2162160,  57657600, ... A001761
%e A124320 [4] 1,  5,  42,  504,  7920,  154440,  3603600,  98017920, ... A102693
%e A124320 [5] 1,  6,  56,  720, 11880,  240240,  5765760, 160392960, ... A093197
%e A124320 [6] 1,  7,  72,  990, 17160,  360360,  8910720, 253955520, ... A203473
%e A124320 [7] 1,  8,  90, 1320, 24024,  524160, 13366080, 390700800, ...
%e A124320 [8] 1,  9, 110, 1716, 32760,  742560, 19535040, 586051200, ...
%e A124320 [9] 1, 10, 132, 2184, 43680, 1028160, 27907200, 859541760, ...
%p A124320 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
%p A124320 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
%t A124320 Table[Pochhammer[n,k], {n,0,5},{k,0,n}]//Flatten (* _Peter Luschny_, Jan 09 2011 *)
%o A124320 (Sage)
%o A124320 for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # _Peter Luschny_, Jan 09 2011
%o A124320 (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
%Y A124320 Rows of array: A000407, A001813, A006963, A001761, A102693, A093197, A203473.
%Y A124320 Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).
%K A124320 nonn,tabl
%O A124320 0,5
%A A124320 _Emeric Deutsch_, Oct 26 2006