A355360 -id:A355360 - cp's OEIS frontend

A355361 G.f. A(x) satisfies: xA(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

1, 1, 5, 26, 136, 746, 4261, 25173, 152596, 943804, 5931561, 37768700, 243124702, 1579577423, 10344340396, 68212177180, 452531832109, 3018280278965, 20227324602249, 136135295125566, 919757424512780, 6235752585125348, 42411283395662960, 289289349007740037

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

Equals the row sums of triangle A355360: a(n) = Sum_{k=0..n} A355360(n,k) for n >= 0.

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 136*x^4 + 746*x^5 + 4261*x^6 + 25173*x^7 + 152596*x^8 + 943804*x^9 + 5931561*x^10 + ...
where
x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355362, A355363, A355364, A355365.

(* Calculation of constants {d,c}: *) {1/r, s*Sqrt[((-1 + s)*(-1 + r*s) * Log[r]*((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s) * Log[r] *  Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*Log[r]*QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s)*((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] - Log[r]*(s + r*(-1 + s) * Derivative[0, 1][QPochhammer][s, r] / QPochhammer[s, r])))) / (2* Pi*(-s*(1 + r - 4*r*s + r*(1 + r)*s^2) * Log[r]^2 + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, -Log[r*s]/Log[r], r]))]} /. FindRoot[{QPochhammer[r] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s)*(-1 + r*s)) == -1, 1/(-1 + s) + 1/(-1 + r*s) + (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/(r*s)]/Log[r], r])/Log[r] == -2}, {r, 1/7}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)

{a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 7.27539777267340429262199058476266... and c = 0.504162727798216251681995853318... - Vaclav Kotesovec, Jul 20 2022
A(1/d) = 1.78721033673795569... where 1/d = 0.1374495293928845... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 30 2022
Formula (3) can be rewritten as the functional equation QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) = x*y / QPochhammer(x). - Vaclav Kotesovec, Jan 19 2024

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+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.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22, 0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140, 0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969, 0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084, 0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820, 0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732

Offset: 0

Paul D. Hanna, Aug 09 2022

T(n, 3*n+1) = [x^n*y^(3*n+1)] A(x,y) = binomial(4*n, n)/(3*n + 1) = A002293(n) for n >= 0, where g.f. G(x) of A002293 satisfies: G(x) = 1 + x*G(x)^4.
T(4*n, 1) = A000716(n) for n >= 0 (nonzero terms in column 1).
T(4*n+3, 2) = [x^(4*n+3)*y^2] A(x,y) = 2 * A354655(n+1) for n >= 0, where A354655 equals column 2 of triangle A354650.
T(4*n+2, 3) = [x^(4*n+2)*y^3] A(x,y) = 4 * A354656(n+1) for n >= 0, where A354656 equals column 3 of triangle A354650.
T(2*n, 2*n+1) = A356504(n), for n >= 0.
T(2*n+1, 2*n) = A356505(n) for n >= 0.
T(3*n, n+1) = A356506(n) for n >= 0.
T(3*n+1, n) = A355365(n) where A355365 is the central terms of A355360 = A355360(2*n,n).
Sum_{k=0..3*n+1} T(n, k) = A354248(n) for n >= 0 (row sums).
Sum_{k=0..3*n+1} T(n, k) * 2^k = A356502(n) for n >= 0.
Sum_{k=0..3*n+1} T(n, k) * 3^k = A356503(n) for n >= 0.
Sum_{k=0..3*n+1} T(4*n+1-k, k) = A355872(n+1) for n >= 0 (nonzero antidiagonal sums).
SPECIFIC VALUES.
(V.1) 1 = A(x,y) at x = -exp(-Pi) and y = Pi^(1/4)/gamma(3/4).
(V.2) 1 = A(x,y) at x = -exp(-2*Pi) and y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) 1 = A(x,y) at x = -exp(-3*Pi) and y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) 1 = A(x,y) at x = -exp(-4*Pi) and y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) 1 = A(x,y) at x = -exp(-sqrt(3)*Pi) and y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			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 irregular triangle of coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
  n = 0: [0, 1];
  n = 1: [1, 0, 0, 0, 1];
  n = 2: [0, 0, 0, 4, 0, 0, 0, 4];
  n = 3: [0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22];
  n = 4: [0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140];
  n = 5: [0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969];
  n = 6: [0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084];
  n = 7: [0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820];
  n = 8: [0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732];
  n = 9: [0, 0, 0, 0, 3250, 0, 0, 0, 160965, 0, 0, 0, 1670942, 0, 0, 0, 6640711, 0, 0, 0, 12167001, 0, 0, 0, 10399545, 0, 0, 0, 3362260];
  ...
Reading this triangle by nonzero antidiagonals [x^(4*n+1-k)*y^k] A(x,y) for n >= 0, k = 0..3*n+1, yields triangle A356501:
  [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];
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..3926

Cf. A354248, A356502, A356503, A356504, A356505, A355872.
Cf. A354655, A354656, A355365, A000716, A002293.
Cf. related tables: A356501, A355350, A355360, A355870.

{T(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,9, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

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.

A355362 G.f. A(x) satisfies: 2xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 2, 14, 106, 852, 7286, 65216, 603714, 5731930, 55506348, 546091942, 5443033448, 54845812094, 557774491672, 5717718435034, 59017814463718, 612873311614338, 6398538141213916, 67121038262747380, 707114126290890810, 7478082640450505012, 79360375914717108922

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 106*x^3 + 852*x^4 + 7286*x^5 + 65216*x^6 + 603714*x^7 + 5731930*x^8 + 55506348*x^9 + 546091942*x^10 + ...
where
2*x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355363, A355364, A355365.

{a(n) = my(A=[1,2],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 2*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} A355360(n,k) * 2^k for n >= 0.
G.f. A(x) satisfies:
(1) 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -2*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) 2*x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 11.38813717738101179115221618346020026348459... and c = 0.5257715220992591718905720654742321646... - Vaclav Kotesovec, Jul 03 2025

A355363 G.f. A(x) satisfies: 3xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 3, 27, 270, 2928, 33912, 411345, 5159337, 66364326, 870637086, 11604385575, 156697653654, 2139109221960, 29472597414681, 409312118499336, 5723853297702444, 80528723782556475, 1139033786793330429, 16187921479930951917, 231046413762053945958

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

			G.f.: A(x) = 1 + 3*x + 27*x^2 + 270*x^3 + 2928*x^4 + 33912*x^5 + 411345*x^6 + 5159337*x^7 + 66364326*x^8 + 870637086*x^9 + 11604385575*x^10 + ...
where
3*x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355362, A355364, A355365.

{a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 3*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} A355360(n,k) * 3^k for n >= 0.
G.f. A(x) satisfies:
(1) 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -3*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) 3*x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 15.42894386025000237511183711088501557092135179... and c = 0.53592940996364915517082259731565361731654... - Vaclav Kotesovec, Jul 03 2025

A355364 G.f. A(x) satisfies: x^2A(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 0, 1, 3, 11, 34, 110, 350, 1147, 3800, 12836, 43929, 152285, 533205, 1883187, 6698612, 23974179, 86258459, 311811314, 1131863444, 4124127216, 15078422405, 55301519095, 203405409935, 750122683729, 2773048061073, 10274442343829, 38147288401915

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

Equals the antidiagonal sums of A355360; a(n) = Sum_{k=0..n} A355360(n-k,k).

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 11*x^4 + 34*x^5 + 110*x^6 + 350*x^7 + 1147*x^8 + 3800*x^9 + 12836*x^10 + 43929*x^11 + 152285*x^12 + ...
where
x^2*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355362, A355363, A355365.

{a(n) = my(A=[1,0],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x^2*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x^2*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x^2*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x^2*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 3.92217771004386918... and c = 0.52890084997249... - Vaclav Kotesovec, Jul 03 2025

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42

Offset: 0

Paul D. Hanna, Jul 19 2022

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Paul D. Hanna, Table of n, a(n) for n = 0..3875

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

{T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

cp's OEIS Frontend

A355365 Central terms of A355360; a(n) = A355360(2*n,n).

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A355361 G.f. A(x) satisfies: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A356500 Coefficients T(n,k) of x^n*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.

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A355362 G.f. A(x) satisfies: 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355363 G.f. A(x) satisfies: 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355364 G.f. A(x) satisfies: x^2*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

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A355361 G.f. A(x) satisfies: xA(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+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.

A355362 G.f. A(x) satisfies: 2xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355363 G.f. A(x) satisfies: 3xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355364 G.f. A(x) satisfies: x^2A(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.