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.

1, 1, 0, 3, 6, 4, 1, 0, 9, 54, 120, 135, 84, 28, 4, 0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22, 0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140, 0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969

Offset: 0

Author

Paul D. Hanna, Aug 09 2022

Keywords

• nonn
• tabf

Comments

Row sums equal A355872.

Alternating row sums equals zero for all rows.

Rightmost border equals [x^n*y^(3*n+1)] A(x,y) = A002293(n) = binomial(4*n, n)/(3*n + 1).

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).

Examples

			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) + ...
such that A = A(x,y) satisfies
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) + ...
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:
n = 0: [1, 1];
n = 1: [0, 3, 6, 4, 1];
n = 2: [0, 9, 54, 120, 135, 84, 28, 4];
n = 3: [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
n = 4: [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
n = 5: [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
n = 6: [0, 221, 14952, 282948, 2578147, 14039250, 51126740, 133101836, 258436719, 384735141, 446971668, 409367712, 296679006, 169713208, 75904032, 26050408, 6640711, 1187433, 133266, 7084];
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Links

• Paul D. Hanna, Table of n, a(n) for n = 0..671

Crossrefs

Cf. A355872 (row sums), A356500 (main entry), A002293 (right border), A000716 (column 1).

Programs

PARI

{T(n,k) = my(A=[y],M); for(i=1,4*n+1, A = concat(A,0); M = ceil(sqrt(4*n+1));
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)}
for(n=0,7, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:

(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).

(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.

(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.

(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).

(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.