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 A361050 #21 Jan 05 2024 23:33:12
%S A361050 1,0,1,0,5,4,0,18,40,22,0,55,244,335,140,0,149,1160,2924,2875,969,0,
%T A361050 371,4688,19090,32745,25081,7084,0,867,16848,103110,272250,352814,
%U A361050 221397,53820,0,1923,55332,485356,1839075,3565548,3709244,1971775,420732,0,4086,169048,2054520,10674985,28909300,44146487,38344384,17682895,3362260
%N A361050 Expansion of g.f. A(x,y) satisfying y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)), as a triangle read by rows.
%C A361050 A359921(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
%C A361050 A359924(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
%C A361050 A361051(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
%C A361050 A361052(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
%C A361050 A361538(n) = T(2*n-1,n-1) for n >= 1.
%C A361050 A360191(n) = T(n+2,1) for n >= 0.
%C A361050 A361535(n) = T(n+3,2)/4 for n >= 0.
%C A361050 A002293(n) = T(n+1,n) for n >= 0.
%H A361050 Paul D. Hanna, <a href="/A361050/b361050.txt">Table of n, a(n) for n = 1..1275</a>
%H A361050 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%F A361050 G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following.
%F A361050 (1) y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)).
%F A361050 (2) y/x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^(2*n-1)*A(x,y)^2) * (1 - x^(2*n-1)/A(x,y)^2), by the Watson quintuple product identity.
%F A361050 (3) Sum_{n>=0} T(n+2,1) * x^n = 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2, which is the g.f. of A360191.
%F A361050 (4) Sum_{n>=0} T(n+3,2) * x^n = 4*F(x) where F(x) = 1/Product_{n>=1} (1 - x^n)^6 * (1 - x^(2*n-1))^4, which is the g.f. of A361535.
%F A361050 (5) Sum_{n>=0} T(n+1,n) * x^n = D(x) where D(x) = 1 + x*D(x)^4 is the g.f. of A002293.
%F A361050 (6) T(n+1,n) = binomial(4*n, n)/(3*n + 1) for n >= 0.
%e A361050 G.f.: A(x,y) = x + y*x^2 + (5*y + 4*y^2)*x^3 + (18*y + 40*y^2 + 22*y^3)*x^4 + (55*y + 244*y^2 + 335*y^3 + 140*y^4)*x^5 + (149*y + 1160*y^2 + 2924*y^3 + 2875*y^4 + 969*y^5)*x^6 + (371*y + 4688*y^2 + 19090*y^3 + 32745*y^4 + 25081*y^5 + 7084*y^6)*x^7 + (867*y + 16848*y^2 + 103110*y^3 + 272250*y^4 + 352814*y^5 + 221397*y^6 + 53820*y^7)*x^8 + (1923*y + 55332*y^2 + 485356*y^3 + 1839075*y^4 + 3565548*y^5 + 3709244*y^6 + 1971775*y^7 + 420732*y^8)*x^9 + (4086*y + 169048*y^2 + 2054520*y^3 + 10674985*y^4 + 28909300*y^5 + 44146487*y^6 + 38344384*y^7 + 17682895*y^8 + 3362260*y^9)*x^10 + ...
%e A361050 This triangle of coefficients T(n,k) of x^n*y^k, n >= 1, k = 0..n-1, in g.f. A(x,y) begins:
%e A361050 1;
%e A361050 0, 1;
%e A361050 0, 5, 4;
%e A361050 0, 18, 40, 22;
%e A361050 0, 55, 244, 335, 140;
%e A361050 0, 149, 1160, 2924, 2875, 969;
%e A361050 0, 371, 4688, 19090, 32745, 25081, 7084;
%e A361050 0, 867, 16848, 103110, 272250, 352814, 221397, 53820;
%e A361050 0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732;
%e A361050 0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260;
%e A361050 0, 8374, 486500, 7984667, 55085875, 199363606, 417661860, 525322468, 391561335, 159463876, 27343888;
%e A361050 0, 16634, 1331056, 28909580, 258486830, 1211896230, 3335033317, 5680806120, 6069336891, 3961602925, 1444601027, 225568798;
%e A361050 ...
%o A361050 (PARI) {T(n,k) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
%o A361050 A[#A] = polcoeff(y/x - prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-4) );
%o A361050 polcoeff(polcoeff(H=Ser(A),n,x),k,y)}
%o A361050 for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))
%Y A361050 Cf. A360191 (column 1), A361535 (column 2), A002293 (diagonal), A361538 (central terms).
%Y A361050 Cf. A359921 (y=1), A359924 (y=2), A361051 (y=3), A361052 (y=4).
%Y A361050 Cf. A002293, A356500 (related table), A361550 (related triangle).
%K A361050 nonn,tabl
%O A361050 1,5
%A A361050 _Paul D. Hanna_, Mar 18 2023