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Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = 12*Product_{j=3..n} ( 12 * binomial(j+2, 4) ) with c(0) = c(1) = 1, read by rows.
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%I #12 Oct 12 2024 21:14:51

%S 1,1,1,1,12,1,1,60,60,1,1,180,900,180,1,1,420,6300,6300,420,1,1,840,

%T 29400,88200,29400,840,1,1,1512,105840,740880,740880,105840,1512,1,1,

%U 2520,317520,4445280,10372320,4445280,317520,2520,1,1,3960,831600,20956320,97796160,97796160,20956320,831600,3960,1

%N Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = 12*Product_{j=3..n} ( 12 * binomial(j+2, 4) ) with c(0) = c(1) = 1, read by rows.

%H G. C. Greubel, <a href="/A174151/b174151.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = 12*Product_{j=3..n} ( 12 * binomial(j+2, 4) ) with c(0) = c(1) = 1.

%F T(n, k) = (6*k/((k+1)^2*(k+2)))*Product_{j=0..3} binomial(n+j-1, k) with T(n, 0) = T(n, n) = 1. - _G. C. Greubel_, Apr 16 2021

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 12, 1;

%e 1, 60, 60, 1;

%e 1, 180, 900, 180, 1;

%e 1, 420, 6300, 6300, 420, 1;

%e 1, 840, 29400, 88200, 29400, 840, 1;

%e 1, 1512, 105840, 740880, 740880, 105840, 1512, 1;

%e 1, 2520, 317520, 4445280, 10372320, 4445280, 317520, 2520, 1;

%e 1, 3960, 831600, 20956320, 97796160, 97796160, 20956320, 831600, 3960, 1;

%t (* First program *)

%t c[n_]:= If[n<2, 1, 12*Product[i*(i^2-1)*(i+2)/2], {i,3,n}]];

%t T[n_, m_] = c[n]/(c[m]*c[n-m]);

%t Table[T[n, m], {n, 0, 10}, {m,0,n}]//Flatten

%t (* Second program *)

%t T[n_, k_]:= If[k==0 || k==n, 1, (6*k/((k+1)^2*(k+2)))*Product[Binomial[n+j-1,k], {j,0,3}] ];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 16 2021 *)

%o (Magma)

%o function T(n,k)

%o if k eq 0 or k eq n then return 1;

%o else return (6*k/((k+1)^2*(k+2)))*(&*[Binomial(n+j-1,k): j in [0..3]]);

%o end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 16 2021

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0 or k==n): return 1

%o else: return (6*k/((k+1)^2*(k+2)))*product(binomial(n+j-1,k) for j in (0..3))

%o flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 16 2021

%Y Cf. A174150.

%K nonn,tabl,easy,less

%O 0,5

%A _Roger L. Bagula_, Mar 10 2010

%E Edited by _G. C. Greubel_, Apr 16 2021