This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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