A156883 - OEIS

%I #13 Jun 14 2021 10:03:57

%S 1,1,1,1,1,2,1,1,9,6,1,1,55,657,24,1,1,193,163405,384345,120,1,1,501,

%T 7152001,26215881175,1799118945,720,1,1,1081,125501001,50886093754945,

%U 227121050616681925,67375205371305,5040,1,1,2059,1262046961,15719063251251501,69513937650491307135745,106253703835242139200091375,20185139902805378865,40320

%N Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^4 - (k+1)^3)^i ) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156883/b156883.txt">Antidiagonal rows n = 0..25, flattened</a>

%F T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^4 - (k+1)^3)^i ) with T(n, 0) = n! (square array).

%F T(n, k) = ( Product_{j=1..n} (k^j*(k+1)^(3*j) -1) )/(k*(k+1)^3 -1)^n with T(n, 0) = n! (square array).

%e Square array begins as:

%e    1,      1,           1,              1, ...;

%e    1,      1,           1,              1, ...;

%e    2,      9,          55,            193, ...;

%e    6,    657,      163405,        7152001, ...;

%e   24, 384345, 26215881175, 50886093754945, ...;

%e Triangle begins as:

%e   1;

%e   1, 1;

%e   1, 1,    2;

%e   1, 1,    9,         6;

%e   1, 1,   55,       657,             24;

%e   1, 1,  193,    163405,         384345,                120;

%e   1, 1,  501,   7152001,    26215881175,         1799118945,            720;

%e   1, 1, 1081, 125501001, 50886093754945, 227121050616681925, 67375205371305, 5040;

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1)^3 + (m+1)^4)^i, {i,0,k-1}], {k,n}]];

%t Table[T[k,n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 12 2021 *)

%t (* Second program *)

%t T[n_, k_]= If[k==0, n!, Product[(k^j*(k+1)^(3*j) -1), {j,n}]/(k*(k+1)^3 -1)^n];

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 14 2021 *)

%o (Sage)

%o def A156883(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^(3*j) -1) for j in (1..n))/(k*(k+1)^3 -1)^n

%o flatten([[A156883(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021

%Y Cf. A156881, A156882, A156885, A156888, A156889.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 17 2009

%E Edited by _G. C. Greubel_, Jun 14 2021