cp's OEIS Frontend

A126445 Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0.

%I A126445 #12 Feb 19 2022 11:50:58
%S A126445 1,1,1,6,3,1,120,36,6,1,4845,969,120,10,1,324632,46376,4495,300,15,1,
%T A126445 32468436,3478761,270725,15180,630,21,1,4529365776,377447148,24040016,
%U A126445 1150626,41664,1176,28,1,840261910995,56017460733,2967205528,122391522,3921225,98770,2016,36,1
%N A126445 Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0.
%C A126445 Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.
%H A126445 G. C. Greubel, <a href="/A126445/b126445.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A126445 T(n,k) = C(n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3!, n-k) for n >= k >= 0.
%e A126445 Formula: T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) is illustrated by:
%e A126445 T(n=4,k=1) = C(C(6,3) - C(3,3), n-k) = C(19,3) = 969;
%e A126445 T(n=4,k=2) = C(C(6,3) - C(4,3), n-k) = C(16,2) = 120;
%e A126445 T(n=5,k=2) = C(C(7,3) - C(4,3), n-k) = C(31,3) = 4495.
%e A126445 Triangle begins:
%e A126445            1;
%e A126445            1,         1;
%e A126445            6,         3,        1;
%e A126445          120,        36,        6,       1;
%e A126445         4845,       969,      120,      10,     1;
%e A126445       324632,     46376,     4495,     300,    15,    1;
%e A126445     32468436,   3478761,   270725,   15180,   630,   21,  1;
%e A126445   4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1;
%t A126445 T[n_, k_]:= Binomial[Binomial[n+2,3] - Binomial[k+2,3], n-k];
%t A126445 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 18 2022 *)
%o A126445 (PARI) T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!, n-k)
%o A126445 (Sage)
%o A126445 def A126445(n,k): return binomial(binomial(n+2,3) - binomial(k+2,3), n-k)
%o A126445 flatten([[A126445(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 18 2022
%Y A126445 Columns: A126446, A126447, A126448, A126449 (row sums).
%Y A126445 Variants: A107862, A126450, A126454, A126457.
%K A126445 nonn,tabl
%O A126445 0,4
%A A126445 _Paul D. Hanna_, Dec 27 2006