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 A121437 #17 May 05 2025 02:59:35
%S A121437 1,-3,1,6,-4,1,-16,14,-6,1,63,-62,33,-9,1,-351,365,-215,72,-13,1,2609,
%T A121437 -2790,1731,-642,143,-18,1,-24636,26749,-17076,6696,-1664,261,-24,1,
%U A121437 284631,-311769,202356,-81963,21684,-3831,444,-31,1,-3909926,4305579,-2822991,1166310,-320515,60768,-8012,713,-39,1
%N A121437 Matrix inverse of triangle A122177, where A122177(n,k) = C( k*(k+1)/2 + n-k + 2, n-k) for n>=k>=0.
%F A121437 (1) T(n,k) = A121436(n-1,k) - A121436(n-1,k+1).
%F A121437 (2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 3)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 3).
%F A121437 G.f.s for column k:
%F A121437 (3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 3);
%F A121437 (4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 3).
%F A121437 From _Benedict W. J. Irwin_, Nov 26 2016: (Start)
%F A121437 Conjecture: The sequence (column 2 of triangle) 14, -62, 365, -2790, 26749, ... is described by a series of nested sums:
%F A121437 14 = Sum_{i=1..4} (i+1),
%F A121437 -62 = -Sum_{i=1..4} (Sum_{j=1..i+1} (j+2)),
%F A121437 365 = Sum_{i=1..4} (Sum_{j=1..i+1} (Sum_{k=1..j+2} (k+3))),
%F A121437 -2790 = -Sum_{i=1..4} (Sum_{j=1..i+1} (Sum_{k=1..j+2} (Sum_{l=1..k+3} (l+4)))). (End)
%e A121437 Triangle begins:
%e A121437 1;
%e A121437 -3, 1;
%e A121437 6, -4, 1;
%e A121437 -16, 14, -6, 1;
%e A121437 63, -62, 33, -9, 1;
%e A121437 -351, 365, -215, 72, -13, 1;
%e A121437 2609, -2790, 1731, -642, 143, -18, 1;
%e A121437 -24636, 26749, -17076, 6696, -1664, 261, -24, 1;
%e A121437 284631, -311769, 202356, -81963, 21684, -3831, 444, -31, 1; ...
%o A121437 (PARI) /* Matrix Inverse of A122177 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r+1,r-c)))); return((M^-1)[n+1,k+1])
%o A121437 (PARI) /* Obtain by g.f. */ T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+3)), n-k)
%Y A121437 Cf. A098568, A107876, A122177, A107885.
%K A121437 sign,tabl
%O A121437 0,2
%A A121437 _Paul D. Hanna_, Aug 27 2006