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A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)*(j+1)*[(k+1)*(k+2)/2 - j*(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.

%I A102400 #10 Jul 21 2024 00:28:20
%S A102400 1,1,1,1,7,7,1,31,139,139,1,127,1567,5711,5711,1,511,15379,126579,
%T A102400 408354,408354,1,2047,143527,2357431,15333661,45605881,45605881,1,
%U A102400 8191,1312219,40769819,473433344,2634441290,7390305396,7390305396,1,32767
%N A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)*(j+1)*[(k+1)*(k+2)/2 - j*(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.
%C A102400 Main diagonal is A082162 (with offset). This sequence is derived from column 0 of A102098.
%e A102400 T(4,2) = 1567 = 1*6 + 31*10 + 139*9
%e A102400 = T(3,0)*R(0,2) + T(3,1)*R(1,2) + T(3,2)*R(2,2).
%e A102400 Rows begin:
%e A102400 [1],
%e A102400 [1,1],
%e A102400 [1,7,7],
%e A102400 [1,31,139,139],
%e A102400 [1,127,1567,5711,5711],
%e A102400 [1,511,15379,126579,408354,408354],
%e A102400 [1,2047,143527,2357431,15333661,45605881,45605881],...
%e A102400 where the transpose of the recurrence coefficients given by
%e A102400 [R^t](n,k) = (k+1)*((n+1)*(n+2)/2 - k*(k+1)/2) form triangle:
%e A102400 [1],
%e A102400 [3,4],
%e A102400 [6,10,9],
%e A102400 [10,18,21,16],
%e A102400 [15,28,36,36,25],...
%e A102400 which equals the matrix square of the triangle:
%e A102400 [1],
%e A102400 [1,2],
%e A102400 [1,2,3],
%e A102400 [1,2,3,4],
%e A102400 [1,2,3,4,5],...
%t A102400 T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[k == 0, 1, If[n == k, T[n, n-1], Sum[T[n-1, j]*(j+1)*((k+1)*(k+2)/2-j*(j+1)/2), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 15 2014, after PARI *)
%o A102400 (PARI) {T(n,k)=if(n<k||k<0,0,if(k==0,1,if(n==k,T(n,n-1), sum(j=0,k,T(n-1,j)*(j+1)*((k+1)*(k+2)/2-j*(j+1)/2)))))}
%Y A102400 Cf. A082162, A102098, A102317.
%K A102400 nonn,tabl
%O A102400 0,5
%A A102400 _Paul D. Hanna_, Jan 06 2005