A356501 -id:A356501 - cp's OEIS frontend

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.

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.

A355872 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

2, 14, 434, 17662, 829314, 42293582, 2276970482, 127359871870, 7328894334338, 431089922960910, 25803242957983410, 1566580082112919422, 96239944539571023362, 5971465584401568096846, 373681955307631772312050, 23556948108319423559281918, 1494606013410312933197468930

Offset: 1

Paul D. Hanna, Aug 09 2022

nonn

Conjecture: a(n) == 2 (mod 4) for n >= 1.
Conjecture: a(2*n-1) == 2 (mod 8) for n >= 1.
Conjecture: a(2*n) == 6 (mod 8) for n >= 1.
Equals the row sums of triangle A356501.

			G.f. A(x) = 2*x + 14*x^5 + 434*x^9 + 17662*x^13 + 829314*x^17 + 42293582*x^21 + 2276970482*x^25 + 127359871870*x^29 + 7328894334338*x^33 + 431089922960910*x^37 + ...
such that A = A(x) satisfies
x = ... + 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) + ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A356500, A356501, A356502, A356503.

(* Calculation of constant d: *) 1/r^4 /. FindRoot[{r*s^4*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[s/r, r^2*s^2]*QPochhammer[r^2*s^2, r^2*s^2] == (r - s)*(-1 + r*s^3), 1/s^3*(3*s + r*(-4 + r*s^3) + 2*r^2*(r - s)*s^2*(-1 + r*s^3)* Derivative[0, 1][QPochhammer][1/(r*s^3), r^2*s^2] / QPochhammer[1/(r*s^3), r^2*s^2] + 2*r^3*s^6*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2] * Derivative[0, 1][QPochhammer][s/r, r^2*s^2] + (r - s)*(-1 + r*s^3)* (-(2*QPolyGamma[0, 1, r^2*s^2] - 3*QPolyGamma[0, Log[1/(r*s^3)]/Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[s/r]/Log[r^2*s^2], r^2*s^2]) / Log[r^2*s^2] + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] /  QPochhammer[r^2*s^2, r^2*s^2])) == 0}, {r, 1/60}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)

{a(n) = my(A=[0,2]); for(i=1,n, A=concat(A,[0,0,0,0]);
A[#A] = -polcoeff( sum(m=-#A,#A,(-x)^(m^2) * Ser(A)^((m-1)^2) ), #A-1)); A[4*n-2]}
for(n=1,20,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3) satisfies:
(1) x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) x = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
(3) -1 = Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 69.7705416198088434764685861402300375255728007801297265... and c = 0.0044667602848752470638241640199049506066862963974858... - Vaclav Kotesovec, Mar 19 2023

cp's OEIS Frontend

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.

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A355872 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

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cp's OEIS Frontend

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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A355872 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

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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.