A203473 -id:A203473 - cp's OEIS frontend

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

Emeric Deutsch, Oct 26 2006

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

			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, ...

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
NIST Digital Library of Mathematical Functions, Pochhammer's Symbol

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

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

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

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

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

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

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

A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

Original entry on oeis.org

1, 7, 504, 498960, 8562153600, 3085457671296000, 27493649380770693120000, 6982164025191299372050022400000, 57286678477842677171688269225656320000000, 16987900892972660430046341200043192304533504000000000, 201504981205067832055356568153709798734509139298353152000000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203470. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n), as in A203474.

G. C. Greubel, Table of n, a(n) for n = 1..35

Cf. A000178, A093883, A203470, A203473, A203474.

[(&*[(&*[i+j: i in [3..j]])/(2*j): j in [3..n+2]]): n in [1..20]]; // G. C. Greubel, Aug 26 2023

a:= n-> mul(mul(i+j, i=3..j-1), j=4..n+2):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= j + 2;    z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}] (* A000178 *)
Table[v[n], {n,z}]             (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]    (* A203473 *)
Table[v[n]/d[n], {n,20}]       (* A203474 *)
(* Second program *)
Table[(18*2^(n+2)^2/Pi^(n/2))*BarnesG[n+3]*BarnesG[n+7/2]/(BarnesG[n+ 6]*BarnesG[7/2]), {n,20}] (* G. C. Greubel, Aug 26 2023 *)

[product( gamma(2*j)/gamma(j+3) for j in range(3,n+3) ) for n in range(1,20)] # G. C. Greubel, Aug 26 2023

a(n) ~ 3*sqrt(A) * 2^(n^2 + 9*n/2 + 185/24) * n^(n^2/2 - n/2 - 179/24) / (Pi^(3/2) * exp(3*n^2/4 - n/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021
From G. C. Greubel, Aug 26 2023: (Start)
a(n) = Prod_{j=3..n+2} Prod_{i=3..j-1} (i + j).
a(n) = Prod_{j=3..n+2} Gamma(2*j)/Gamma(j+3).
a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/(BarnesG(n+ 6)*BarnesG(7/2)). (End)

Name edited by Alois P. Heinz, Jul 23 2017

A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

Original entry on oeis.org

1, 7, 252, 41580, 29729700, 89278289100, 1104908105901600, 55674109640169820800, 11329124570678156834592000, 9258047307912482983660236480000, 30262334718212007877669234596364800000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A093883, A203472, A203473.

[(&*[ Binomial(2*j+3, j+4): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 27 2023

(* First program *)
f[j_]:= j+2; z=16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,n}]  (* A000178(n-1) *)
Table[v[n], {n,z}]              (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]     (* A203473 *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Product[Binomial[2*j+3, j+4], {j,n}], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

[product( binomial(2*j+5,j+5) for j in range(n) ) for n in range(1,20)] # G. C. Greubel, Aug 27 2023

a(n) ~ 3*A^(3/2) * 2^(n^2 + 4*n + 185/24) * exp(n/2 - 1/8) / (Pi^(n/2 + 3/2) * n^(n/2 + 59/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021
From G. C. Greubel, Aug 27 2023: (Start)
a(n) = Product_{j=1..n} binomial(2*j+3, j+4).
a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/( BarnesG(n +1)*BarnesG(n+6)*BarnesG(7/2)). (End)

Definition corrected by Vaclav Kotesovec, Apr 09 2021

cp's OEIS Frontend

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

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A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

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A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

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