A354248 -id:A354248 - 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.

A356502 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

2, 17, 544, 24344, 1261702, 71159152, 4240009152, 262584135640, 16734002688722, 1090225325371424, 72285357987696768, 4861658409827006872, 330874470176939132844, 22744684876060771599568, 1576898258893213475814464, 110136698483814852518084528, 7742091796859524187452564262

Offset: 0

Paul D. Hanna, Aug 09 2022

nonn

			G.f.: A(x) = 2 + 17*x + 544*x^2 + 24344*x^3 + 1261702*x^4 + 71159152*x^5 + 4240009152*x^6 + 262584135640*x^7 + 16734002688722*x^8 + ...
such that A = A(x) satisfies
2 = ... + 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 = 0..300

Cf. A356500, A354248, A356503, A356504.

(* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], 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} /. k -> 2, {r, 1/75}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)

{a(n) = my(A=[2],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)); H=A; A[n+1]}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) 2 = 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) 2 = (-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-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 77.309779325704779292317107617559471210592218708634855530355675234... and c = 0.31219183409397424726366930735250286274022579073644627976468... - Vaclav Kotesovec, Mar 19 2023

A356503 G.f. A(x) satisfies: 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

3, 82, 8856, 1319544, 227536218, 42679033812, 8455886664768, 1741107313315440, 368888770098828828, 79897573332771325074, 17610753240158104125072, 3937441977622780631428392, 890818276864624495645873656, 203562312272030478854160019188, 46914726894168080421554447339136

Offset: 0

Paul D. Hanna, Aug 09 2022

nonn

			G.f.: A(x) = 3 + 82*x + 8856*x^2 + 1319544*x^3 + 227536218*x^4 + 42679033812*x^5 + 8455886664768*x^6 + 1741107313315440*x^7 + 368888770098828828*x^8 + ...
such that A = A(x) satisfies
3 = ... + 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 = 0..300

Cf. A356500, A354248, A356502.

(* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], 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} /. k -> 3, {r, 1/250}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)

{a(n) = my(A=[3],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)); H=A; A[n+1]}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) 3 = 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) 3 = (-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-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 256.98186313678886207367060740797984009618789137465012036784040337275... and c = 0.462191657806050531963307936402124127070124175540050112607444087... - Vaclav Kotesovec, Mar 19 2023

A355349 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n(n-1)/2) A(x)^(n^2).

Original entry on oeis.org

1, 2, 10, 76, 678, 6608, 68170, 731638, 8084692, 91361298, 1050937008, 12264790410, 144856757032, 1728197200206, 20796217437806, 252117655811806, 3076371017010508, 37753163861001044, 465657991700212170, 5769586313420410060, 71777257553636752194

Offset: 0

Paul D. Hanna, Aug 02 2022

nonn

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 76*x^3 + 678*x^4 + 6608*x^5 + 68170*x^6 + 731638*x^7 + 8084692*x^8 + 91361298*x^9 + 1050937008*x^10 + ...
where
2 = ... + x^6*A(x)^9 - x^3*A(x)^4 - x*A(x) + A(x) + 1 - x*A(x) - x^3*A(x)^4 + x^6*A(x)^9 + x^10*A(x)^16 - x^15*A(x)^25 - x^21*A(x)^36 ++-- ... (-x)^(n*(n-1)/2) * A(x)^(n^2) + ...

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

Cf. A355871, A354248, A354662.

(* Calculation of constant d: *) 1/r /. FindRoot[{r*s^2*QPochhammer[-1/s, -r*s^2]* QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2]/((1 + s)*(-1 + r*s)) == 2, (4 + 2*s - 2*r*s)/(r*s^2) + 2*r*s^2*QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2] * Derivative[0, 1][QPochhammer][-1/s, -r*s^2] + 4*(1 + s)*(-1 + r*s)* Derivative[0, 1][QPochhammer][1/(r*s), -r*s^2] / QPochhammer[1/(r*s), -r*s^2] + 2*(1 + s)*(-1 + r*s) * ((2*QPolyGamma[0, 1, -r*s^2] - QPolyGamma[0, Log[-1/s]/Log[-r*s^2], -r*s^2] - QPolyGamma[0, Log[1/(r*s)]/Log[-r*s^2], -r*s^2])/(r*s^2* Log[-r*s^2]) + (2*Derivative[0, 1][QPochhammer][-r*s^2, -r*s^2])/ QPochhammer[-r*s^2]) == 0}, {r, 1/12}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 31 2024 *)

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

G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^2).
(2) 2 = Product_{n>=1} (1 - (-x)^n*A(x)^(2*n)) * (1 + (-x)^(n-1)*A(x)^(2*n-1)) * (1 + (-x)^n*A(x)^(2*n-1)), by the Jacobi triple product identity.
From Vaclav Kotesovec, Jan 31 2024: (Start)
Formula (2) can be rewritten as the functional equation QPochhammer(-x*y^2) * QPochhammer(1/(x*y), -x*y^2)/(1 - 1/(x*y)) * QPochhammer(-1/y, -x*y^2)/(1 + 1/y) = 2.
a(n) ~ c * d^n / n^(3/2), where d = 13.4235502463317299100709807099986120871056759637108569... and c = 0.1810118197770993236368884418746617625188562698029... (End)

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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A356502 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

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A356503 G.f. A(x) satisfies: 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

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A355349 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^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.

A355349 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n(n-1)/2) A(x)^(n^2).