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A357400 Coefficients T(n,k) of x^n*y^k in the function A(x,y) that satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x,y)^n, as a triangle read by rows with k = 0..n for each row index n >= 0.

%I A357400 #16 Oct 08 2022 00:26:10
%S A357400 1,0,1,0,0,2,0,1,0,5,0,0,3,0,14,0,-2,0,10,0,42,0,8,-12,0,35,0,132,0,
%T A357400 -14,36,-52,0,126,0,429,0,16,-76,148,-210,0,462,0,1430,0,-7,84,-354,
%U A357400 590,-825,0,1716,0,4862,0,-24,-27,416,-1565,2322,-3199,0,6435,0,16796,0,103,-276,-120,1950,-6732,9086,-12320,0,24310,0,58786,0,-232,987,-1752,-560,8832,-28490,35464,-47268,0,92378,0,208012
%N A357400 Coefficients T(n,k) of x^n*y^k in the function A(x,y) that satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x,y)^n, as a triangle read by rows with k = 0..n for each row index n >= 0.
%C A357400 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%C A357400 T(n,n) = binomial(2*n+1, n+1)/(2*n+1) = A000108(n) for n >= 0.
%C A357400 T(n+1,n) = 0 for n>= 0.
%C A357400 T(n+2,n) = binomial(2*n-1, n-1) = A001700(n-1) for n >= 1.
%C A357400 T(n+3,n) = 0 for n>= 0.
%C A357400 T(n+1,1) = A357401(n) for n >= 0.
%C A357400 A356783(n) = Sum_{k=0..n} T(n,k), for n >= 0.
%C A357400 A357402(n) = Sum_{k=0..n} T(n,k) * 2^k, for n >= 0.
%C A357400 A357403(n) = Sum_{k=0..n} T(n,k) * 3^k, for n >= 0.
%C A357400 A357404(n) = Sum_{k=0..n} T(n,k) * 4^k, for n >= 0.
%C A357400 A357405(n) = Sum_{k=0..n} T(n,k) * 5^k, for n >= 0.
%H A357400 Paul D. Hanna, <a href="/A357400/b357400.txt">Table of n, a(n) for n = 0..5150</a>
%F A357400 G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies the following relations.
%F A357400 (1) y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x,y)^n.
%F A357400 (2) y*x*A(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x,y)^n ).
%F A357400 (3) -y*x*A(x,y)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x,y)^n / (1 - x^(n+1)*A(x,y))^n.
%F A357400 (4) -y*A(x,y)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x,y) - x^n)^(n+1) / A(x)^n.
%F A357400 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x,y))^(n+1) / A(x,y)^n.
%F A357400 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x,y)^n / (A(x,y) - x^(n+1))^n.
%e A357400 G.f. A(x,y) = 1 + x*y + x^2*(2*y^2) + x^3*(y + 5*y^3) + x^4*(3*y^2 + 14*y^4) + x^5*(-2*y + 10*y^3 + 42*y^5) + x^6*(8*y - 12*y^2 + 35*y^4 + 132*y^6) + x^7*(-14*y + 36*y^2 - 52*y^3 + 126*y^5 + 429*y^7) + x^8*(16*y - 76*y^2 + 148*y^3 - 210*y^4 + 462*y^6 + 1430*y^8) + x^9*(-7*y + 84*y^2 - 354*y^3 + 590*y^4 - 825*y^5 + 1716*y^7 + 4862*y^9) + x^10*(-24*y - 27*y^2 + 416*y^3 - 1565*y^4 + 2322*y^5 - 3199*y^6 + 6435*y^8 + 16796*y^10) + ...
%e A357400 such that
%e A357400 y = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x,y)^2 + x^(-1)/A(x,y) + x*0 + x^3*(1 - x)^2*A(x,y) + x^5*(1 - x^2)^3*A(x,y)^2 + x^7*(1 - x^3)^4*A(x,y)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x,y)^n + ...
%e A357400 also
%e A357400 -y*A(x,y)^3 = ... + x^(-3)*(A(x,y) - x^(-2))^(-1)*A(x,y)^2 + x^(-1)*A(x,y) + x*(A(x,y) - 1) + x^3*(A(x,y) - x)^2/A(x,y) + x^5*(1 - x^2)^3/A(x,y)^2 + x^7*(A(x,y) - x^3)^4/A(x,y)^3 + ... + x^(2*n+1)*(A(x,y) - x^n)^(n+1)/A(x,y)^n + ...
%e A357400 This triangle of coefficients T(n,k) of x^n*y^k, k = 0..n, in g.f. A(x,y) begins:
%e A357400 n = 0: [1],
%e A357400 n = 1: [0, 1],
%e A357400 n = 2: [0, 0, 2],
%e A357400 n = 3: [0, 1, 0, 5],
%e A357400 n = 4: [0, 0, 3, 0, 14],
%e A357400 n = 5: [0, -2, 0, 10, 0, 42],
%e A357400 n = 6: [0, 8, -12, 0, 35, 0, 132],
%e A357400 n = 7: [0, -14, 36, -52, 0, 126, 0, 429],
%e A357400 n = 8: [0, 16, -76, 148, -210, 0, 462, 0, 1430],
%e A357400 n = 9: [0, -7, 84, -354, 590, -825, 0, 1716, 0, 4862],
%e A357400 n = 10: [0, -24, -27, 416, -1565, 2322, -3199, 0, 6435, 0, 16796],
%e A357400 n = 11: [0, 103, -276, -120, 1950, -6732, 9086, -12320, 0, 24310, 0, 58786],
%e A357400 n = 12: [0, -232, 987, -1752, -560, 8832, -28490, 35464, -47268, 0, 92378, 0, 208012],
%e A357400 n = 13: [0, 334, -2160, 6436, -9460, -2673, 39102, -119296, 138294, -180960, 0, 352716, 0, 742900],
%e A357400 n = 14: [0, -256, 3002, -14484, 36218, -46902, -12929, 170368, -495846, 539240, -691900, 0, 1352078, 0, 2674440], ...
%e A357400 in which the main diagonal equals the Catalan numbers (A000108).
%o A357400 (PARI) {T(n,k) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357400 A[#A] = polcoeff(y - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); polcoeff(A[n+1],k,y)}
%o A357400 for(n=0, 15, for(k=0,n, print1(T(n,k), ", "));print(""))
%Y A357400 Cf. A356783 (row sums), A357402 (y=2), A357403 (y=3), A357404 (y=4), A357405 (y=5).
%Y A357400 Cf. A357401 (column 1), A357151, A000108, A001700.
%K A357400 sign,tabl
%O A357400 0,6
%A A357400 _Paul D. Hanna_, Sep 26 2022