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 A215241 #16 Sep 05 2023 22:22:25
%S A215241 1,1,1,3,4,1,18,26,9,1,172,256,99,16,1,2313,3489,1416,264,25,1,40626,
%T A215241 61696,25650,5120,575,36,1,887326,1352518,569772,117980,14450,1098,49,
%U A215241 1,23282964,35566368,15099042,3193728,410850,34608,1911,64,1,715540140,1094499820,466865280,100049120,13259705,1186857,73696,3104,81,1
%N A215241 Unsigned matrix inverse of triangle A214398, as a triangle read by rows n >= 1.
%H A215241 Paul D. Hanna, <a href="/A215241/b215241.txt">Table of n, a(n) for n = 1..1081</a>
%F A215241 G.f.: x*y/(1-x*y) = Sum_{n>=1} Sum_{k=1..n} T(n,k)*x^n*y^k/(1+x)^(n^2).
%F A215241 G.f. of column k: 1 = Sum_{n>=k} T(n,k)*x^(n-k)/(1+x)^(n^2).
%F A215241 Column 1 forms A177447.
%F A215241 Row sums form A133316.
%e A215241 Triangle begins:
%e A215241 1;
%e A215241 1, 1;
%e A215241 3, 4, 1;
%e A215241 18, 26, 9, 1;
%e A215241 172, 256, 99, 16, 1;
%e A215241 2313, 3489, 1416, 264, 25, 1;
%e A215241 40626, 61696, 25650, 5120, 575, 36, 1;
%e A215241 887326, 1352518, 569772, 117980, 14450, 1098, 49, 1;
%e A215241 23282964, 35566368, 15099042, 3193728, 410850, 34608, 1911, 64, 1;
%e A215241 ...
%e A215241 The matrix inverse is a signed version of triangle A214398:
%e A215241 1;
%e A215241 -1, 1;
%e A215241 1, -4, 1;
%e A215241 -1, 10, -9, 1;
%e A215241 1, -20, 45, -16, 1;
%e A215241 -1, 35, -165, 136, -25, 1;
%e A215241 1, -56, 495, -816, 325, -36, 1;
%e A215241 -1, 84, -1287, 3876, -2925, 666, -49, 1; ...
%e A215241 in which the g.f. of column k is 1/(1+x)^(k^2) for k >= 1.
%e A215241 ILLUSTRATE G.F. OF COLUMNS:
%e A215241 k=1: 1 = 1/(1+x) + 1*x/(1+x)^4 + 3*x^2/(1+x)^9 + 18*x^3/(1+x)^16 + 172*x^4/(1+x)^25 + 2313*x^5/(1+x)^36 + 40626*x^6/(1+x)^49 + ...
%e A215241 k=2: 1 = 1/(1+x)^4 + 4*x/(1+x)^9 + 26*x^2/(1+x)^16 + 256*x^3/(1+x)^25 + 3489*x^4/(1+x)^36 + 61696*x^5/(1+x)^49 + ...
%e A215241 k=3: 1 = 1/(1+x)^9 + 9*x/(1+x)^16 + 99*x^2/(1+x)^25 + 1416*x^3/(1+x)^36 + 25650*x^4/(1+x)^49 + ...
%e A215241 k=4: 1 = 1/(1+x)^16 + 16*x/(1+x)^25 + 264*x^2/(1+x)^36 + 5120*x^3/(1+x)^49 + ...
%t A215241 T[n_, k_] := Module[{M}, M = Table[Binomial[c^2 + r - c - 1, r - c], {r, 1, n}, {c, 1, n}]; (-1)^(n - k) Inverse[M][[n, k]]];
%t A215241 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 05 2023, after PARI program *)
%o A215241 (PARI) {T(n, k)=local(M=matrix(n,n,r,c,binomial(c^2+r-c-1, r-c)));(-1)^(n-k)*(M^-1)[n,k]}
%o A215241 for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))
%Y A215241 Cf. A177447 (column 1), A215242 (column 2), A215243 (column 3); A133316 (row sums).
%Y A215241 Cf. A214398 (unsigned matrix inverse).
%K A215241 nonn,tabl
%O A215241 1,4
%A A215241 _Paul D. Hanna_, Aug 06 2012