A359920 -id:A359920 - cp's OEIS frontend

A361555 Expansion of g.f. A(x) satisfying 5x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).

1, 5, 50, 465, 4925, 59870, 776155, 10364135, 142082065, 1995371980, 28549274995, 414327073520, 6084353526535, 90258375062245, 1350607531232830, 20361436162127965, 308964002231172075, 4715119823819824535, 72324133311820587435, 1114404268419043050750

Offset: 0

Paul D. Hanna, Mar 19 2023

nonn

			G.f.: A(x) = 1 + 5*x + 50*x^2 + 465*x^3 + 4925*x^4 + 59870*x^5 + 776155*x^6 + 10364135*x^7 + 142082065*x^8 + 1995371980*x^9 + ...
where A = A(x) satisfies the doubly infinite sum
5*x = ... + x^12*(1/A^9 - A^8) + x^5*(1/A^6 - A^5) + x*(1/A^3 - A^2) + (1 - 1/A) + x^2*(A^3 - 1/A^4) + x^7*(A^6 - 1/A^7) + x^15*(A^9 - 1/A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...
also, by the Watson quintuple product identity,
5*x = (1-x)*(1-x*A)*(1-1/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1-x^2*A)*(1-x/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^2/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^3/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Cf. A361550, A359920, A361552, A361553, A361554.

(* Calculation of constant d: *) 1/r /. FindRoot[{r^2 * s^3 * QPochhammer[r] * QPochhammer[1/(r*s^2), r^2] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] *(QPochhammer[s^2/r, r^2]/ ((-1 + s)*(-1 + r*s)*(-r + s^2)*(-1 + r*s^2))) == 5, (3*r - 2*r*(1 + r)*s - s^2 + r^3*s^4 + 2*r*(1 + r)*s^5 - 3*r^2*s^6)*Log[r] + (-1 + s)*(-1 + r*s)*(r - s^2)*(-1 + r*s^2) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -1/2 - Log[s]/Log[r], r^2] + QPolyGamma[0, -1/2 + Log[s]/Log[r], r^2] - QPolyGamma[0, -Log[r*s]/Log[r], r]) == 0}, {r, 1/16}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)

/* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(4*x - sum(m=-#A, #A, x^(m*(3*m-1)/2) * Ser(A)^(3*m-1) * (x^m*Ser(A) - 1) ) , #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

/* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(5*x - prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 5*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) 5*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) 5*x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
(4) a(n) = Sum_{k=0..n} A361550(n,k) * 5^k for n >= 0.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (3) can be rewritten as the functional equation 5*x = QPochhammer(x) * QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) * QPochhammer(y^2/x, x^2)/(1-y^2/x) * QPochhammer(1/(x*y^2), x^2)/(1-1/(x*y^2)).
a(n) ~ c * d^n / n^(3/2), where d = 16.695183607901729043700484293708659594719464935528330676878595927048... and c = 0.5534958293134675625595273281664529583363592593727800077222126752653... (End)

A359719 a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (exp(3nx) - exp(-(3n+1)x)).

Original entry on oeis.org

1, -11, 58, -225, 2146, -14821, 85590, -1974433, 9180658, 2927259, -85838114, 63964584095, -520091681238, 16934937109019, -384678052715594, 5238404820228159, -295855770548974622, 4600244140822151099, -186350295911412573810, 4851711966859680480959

Offset: 1

Paul D. Hanna, Jan 22 2023

sign

			E.g.f.: A(x) = x - 11*x^2/2! + 58*x^3/3! - 225*x^4/4! + 2146*x^5/5! - 14821*x^6/6! + 85590*x^7/7! - 1974433*x^8/8! + 9180658*x^9/9! + 2927259*x^10/10! + ...
where A(x) equals the doubly infinite series
A(x) = ... + x^12*(exp(-9*x) - exp(8*x)) + x^5*(exp(-6*x) - exp(5*x)) + x*(exp(-3*x) - exp(2*x)) + (1 - exp(-x)) + x^2*(exp(3*x) - exp(-4*x)) + x^7*(exp(6*x) - exp(-7*x)) + x^15*(exp(9*x) - exp(-10*x)) + ... + x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)) + ...
also, by the Watson quintuple product identity,
A(x) = (1-x)*(1-x*exp(x))*(1-1*exp(-x))*(1-x*exp(2*x))*(1-x*exp(-2*x)) * (1-x^2)*(1-x^2*exp(x))*(1-x*exp(-x))*(1-x^3*exp(2*x))*(1-x^3*exp(-2*x)) * (1-x^3)*(1-x^3*exp(x))*(1-x^2*exp(-x))*(1-x^5*exp(2*x))*(1-x^5*exp(-2*x)) * (1-x^4)*(1-x^4*exp(x))*(1-x^3*exp(-x))*(1-x^7*exp(2*x))*(1-x^7*exp(-2*x)) * ...

Paul D. Hanna, Table of n, a(n) for n = 1..400
Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Cf. A359920, A359919.

/* Using the doubly infinite series */
{a(n) = my(X=x+x*O(x^n),M=sqrtint(2*n)); n! * polcoeff( sum(m=-M,M, x^(m*(3*m+1)/2) * (exp(3*m*X) - exp(-(3*m+1)*X)) ), n)}
for(n=1,30, print1(a(n),", "))

/* Using the quintuple product */
{a(n) = my(X=x+x*O(x^n)); n! * polcoeff( prod(m=1,n, (1 - x^m) * (1 - x^m*exp(X)) * (1 - x^(m-1)*exp(-X)) * (1 - x^(2*m-1)*exp(2*X)) * (1 - x^(2*m-1)*exp(-2*X)) ),n)}
for(n=1,30, print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined by the following.
(1) A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).
(2) A(x) = Product_{n>=1} (1 - x^n) * (1 - x^n*exp(x)) * (1 - x^(n-1)*exp(-x)) * (1 - x^(2*n-1)*exp(2*x)) * (1 - x^(2*n-1)*exp(-2*x)), by the Watson quintuple product identity.
(3) A(x) = 2*exp(-x/2) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * sinh((6*n+1)*x/2).
(4) A(x) = (1 - exp(-x)) * Product_{n>=1} (1 - x^n) * (1 - 2*x^n*cosh(x) + x^(2*n)) * (1 - 2*x^(2*n-1)*cosh(2*x) + x^(4*n-2)).

A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (A(x)^(3n) - 1/A(x)^(3n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).

Original entry on oeis.org

1, 1, 5, 21, 90, 423, 2209, 12261, 69842, 403722, 2367829, 14096616, 85043323, 518567546, 3189349181, 19758783404, 123200215388, 772606927013, 4870004002571, 30837536428981, 196065919894270, 1251190368574657, 8011186350002373, 51451177669973807, 331365405433649972

Offset: 0

Paul D. Hanna, Mar 20 2023

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 90*x^4 + 423*x^5 + 2209*x^6 + 12261*x^7 + 69842*x^8 + 403722*x^9 + 2367829*x^10 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Cf. A359920.

(* Calculation of constant d: *) 1/r /. FindRoot[{r^3*s^3 * QPochhammer[1/(r*s^2), r^2] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] * QPochhammer[s^2/r, r^2] / ((-1 + s)*(-1 + r*s)*(-r + s^2)*(-1 + r*s^2)) == r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + 2*s/(-r + s^2) - 2/(s - r*s^3) + (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -1/2 - Log[s]/Log[r], r^2] + QPolyGamma[0, -1/2 + Log[s]/Log[r], r^2] - QPolyGamma[0, -Log[r*s]/Log[r], r])/(s*Log[r]) == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)

/* Using the doubly infinite series */
{a(n) = my(A=[1],P = 1/prod(m=1,n,1-x^m +x*O(x^n))); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - P*sum(m=-#A, #A, x^(m*(3*m-1)/2) * Ser(A)^(3*m-1) * (x^m*Ser(A) - 1) ) , #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

/* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - prod(m=1, #A, (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n, along with P(x) = 1/Product_{n>=1} (1 - x^n), satisfies the following.
(1) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 6.8614556596748153... and c = 0.3325444675505... - Vaclav Kotesovec, Mar 20 2023

cp's OEIS Frontend

A361555 Expansion of g.f. A(x) satisfying 5x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).

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A359719 a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (exp(3nx) - exp(-(3n+1)x)).

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A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (A(x)^(3n) - 1/A(x)^(3n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).

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cp's OEIS Frontend

A361555 Expansion of g.f. A(x) satisfying 5*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

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A359719 a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).

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A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).

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A361555 Expansion of g.f. A(x) satisfying 5x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).

A359719 a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (exp(3nx) - exp(-(3n+1)x)).

A360580 Expansion of g.f. A(x) satisfying x = P(x) * Sum_{n=-oo..+oo} x^(n(3n+1)/2) * (A(x)^(3n) - 1/A(x)^(3n+1)), where P(x) = 1/Product_{n>=1} (1 - x^n).