cp's OEIS Frontend

A356501 Coefficients T(n,k) of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

%I A356501 #11 Aug 11 2022 07:24:57
%S A356501 1,1,0,3,6,4,1,0,9,54,120,135,84,28,4,0,22,294,1360,3250,4662,4284,
%T A356501 2568,981,219,22,0,51,1260,10120,41405,103020,170324,196172,160965,
%U A356501 94390,38896,10764,1807,140,0,108,4590,58380,368145,1404102,3587696,6515712,8715465,8763645,6684744,3863496,1670942,525980,114240,15368,969
%N A356501 Coefficients T(n,k) of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.
%C A356501 Row sums equal A355872.
%C A356501 Alternating row sums equals zero for all rows.
%C A356501 Rightmost border equals [x^n*y^(3*n+1)] A(x,y) = A002293(n) = binomial(4*n, n)/(3*n + 1).
%C A356501 This triangle may be formed from the nonzero antidiagonals of triangle A356500; see main entry A356500 for further formulas for the coefficients in g.f. A(x,y).
%H A356501 Paul D. Hanna, <a href="/A356501/b356501.txt">Table of n, a(n) for n = 0..671</a>
%F A356501 G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
%F A356501 (1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
%F A356501 (2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
%F A356501 (3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
%F A356501 (4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
%F A356501 (5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.
%e A356501 G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
%e A356501 such that A = A(x,y) satisfies
%e A356501 y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
%e A356501 This triangle of coefficients of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
%e A356501 n = 0: [1, 1];
%e A356501 n = 1: [0, 3, 6, 4, 1];
%e A356501 n = 2: [0, 9, 54, 120, 135, 84, 28, 4];
%e A356501 n = 3: [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
%e A356501 n = 4: [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
%e A356501 n = 5: [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
%e A356501 n = 6: [0, 221, 14952, 282948, 2578147, 14039250, 51126740, 133101836, 258436719, 384735141, 446971668, 409367712, 296679006, 169713208, 75904032, 26050408, 6640711, 1187433, 133266, 7084];
%e A356501 ...
%o A356501 (PARI) {T(n,k) = my(A=[y],M); for(i=1,4*n+1, A = concat(A,0); M = ceil(sqrt(4*n+1));
%o A356501 A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[4*n+2-k],k,y)}
%o A356501 for(n=0,7, for(k=0,3*n+1, print1(T(n,k),", "));print(""))
%Y A356501 Cf. A355872 (row sums), A356500 (main entry), A002293 (right border), A000716 (column 1).
%K A356501 nonn,tabf
%O A356501 0,4
%A A356501 _Paul D. Hanna_, Aug 09 2022