cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A361556 Central terms of triangle A361550.

Original entry on oeis.org

1, 5, 61, 1660, 47460, 1621116, 58002140, 2213389940, 87301563690, 3555890156445, 148125509781095, 6292884402884976, 271565202254735207, 11878392121526009800, 525519782174930309205, 23481280252471520720288, 1058270749214634093475910, 48058678036035725619136698
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

Links

Crossrefs

Cf. A361550.

Programs

  • PARI
    {A361550(n,k) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x*y - 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) );
polcoeff(polcoeff(Ser(A),n,x),k,y)}
for(n=0, 20, print1(A361550(2*n,n), ", "))

Formula

a(n) = A361550(2*n,n) for n >= 0.
a(n) ~ c * d^n / n^2, where d = 51.1751... and c = 0.12624... - Vaclav Kotesovec, Mar 19 2023

A361050 Expansion of g.f. A(x,y) satisfying y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)), as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 5, 4, 0, 18, 40, 22, 0, 55, 244, 335, 140, 0, 149, 1160, 2924, 2875, 969, 0, 371, 4688, 19090, 32745, 25081, 7084, 0, 867, 16848, 103110, 272250, 352814, 221397, 53820, 0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732, 0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260
Offset: 1

Views

Author

Paul D. Hanna, Mar 18 2023

Keywords

Comments

A359921(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A359924(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A361051(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A361052(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A361538(n) = T(2*n-1,n-1) for n >= 1.
A360191(n) = T(n+2,1) for n >= 0.
A361535(n) = T(n+3,2)/4 for n >= 0.
A002293(n) = T(n+1,n) for n >= 0.

Examples

			G.f.: A(x,y) = x + y*x^2 + (5*y + 4*y^2)*x^3 + (18*y + 40*y^2 + 22*y^3)*x^4 + (55*y + 244*y^2 + 335*y^3 + 140*y^4)*x^5 + (149*y + 1160*y^2 + 2924*y^3 + 2875*y^4 + 969*y^5)*x^6 + (371*y + 4688*y^2 + 19090*y^3 + 32745*y^4 + 25081*y^5 + 7084*y^6)*x^7 + (867*y + 16848*y^2 + 103110*y^3 + 272250*y^4 + 352814*y^5 + 221397*y^6 + 53820*y^7)*x^8 + (1923*y + 55332*y^2 + 485356*y^3 + 1839075*y^4 + 3565548*y^5 + 3709244*y^6 + 1971775*y^7 + 420732*y^8)*x^9 + (4086*y + 169048*y^2 + 2054520*y^3 + 10674985*y^4 + 28909300*y^5 + 44146487*y^6 + 38344384*y^7 + 17682895*y^8 + 3362260*y^9)*x^10 + ...
This triangle of coefficients T(n,k) of x^n*y^k, n >= 1, k = 0..n-1, in g.f. A(x,y) begins:
1;
0, 1;
0, 5, 4;
0, 18, 40, 22;
0, 55, 244, 335, 140;
0, 149, 1160, 2924, 2875, 969;
0, 371, 4688, 19090, 32745, 25081, 7084;
0, 867, 16848, 103110, 272250, 352814, 221397, 53820;
0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732;
0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260;
0, 8374, 486500, 7984667, 55085875, 199363606, 417661860, 525322468, 391561335, 159463876, 27343888;
0, 16634, 1331056, 28909580, 258486830, 1211896230, 3335033317, 5680806120, 6069336891, 3961602925, 1444601027, 225568798;
...

Links

Crossrefs

Cf. A360191 (column 1), A361535 (column 2), A002293 (diagonal), A361538 (central terms).
Cf. A359921 (y=1), A359924 (y=2), A361051 (y=3), A361052 (y=4).
Cf. A002293, A356500 (related table), A361550 (related triangle).

Programs

  • PARI
    {T(n,k) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(y/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-4) );
polcoeff(polcoeff(H=Ser(A),n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following.
(1) y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)).
(2) y/x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^(2*n-1)*A(x,y)^2) * (1 - x^(2*n-1)/A(x,y)^2), by the Watson quintuple product identity.
(3) Sum_{n>=0} T(n+2,1) * x^n = 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2, which is the g.f. of A360191.
(4) Sum_{n>=0} T(n+3,2) * x^n = 4*F(x) where F(x) = 1/Product_{n>=1} (1 - x^n)^6 * (1 - x^(2*n-1))^4, which is the g.f. of A361535.
(5) Sum_{n>=0} T(n+1,n) * x^n = D(x) where D(x) = 1 + x*D(x)^4 is the g.f. of A002293.
(6) T(n+1,n) = binomial(4*n, n)/(3*n + 1) for n >= 0.

A361552 Expansion of g.f. A(x) satisfying 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

Original entry on oeis.org

1, 2, 14, 84, 530, 3770, 29446, 240302, 2003914, 17024332, 147306448, 1294859540, 11524690228, 103605031978, 939357512086, 8580744729478, 78898896072996, 729661925134886, 6782435427053490, 63332055630823770, 593793935288453260, 5587934788557993846
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

Comments

Conjecture: a(n) == A359914(n-1) (mod 4) for n > 1.
Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 84*x^3 + 530*x^4 + 3770*x^5 + 29446*x^6 + 240302*x^7 + 2003914*x^8 + 17024332*x^9 + ...
where A = A(x) satisfies the doubly infinite sum
2*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,
2*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) * ...

Links

Crossrefs

Cf. A361550, A359920, A361553, A361554, A361555.
Cf. A359914, A040051, A000041.

Programs

  • Mathematica
    (* Calculation of constant d: *) With[{k = 2}, 1/r /. FindRoot[{r^3*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)) == k*r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + (2*s)/(-r + s^2) - 2/(s - r*s^3) + (-QPolyGamma[0, -Log[r*s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s^2]/Log[r^2], r^2] + QPolyGamma[0, Log[s^2/r]/Log[r^2], r^2]) / (s*Log[r]) == 0}, {r, 1/10}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(2*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), ", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(2*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), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) 2*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) 2*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) * 2^k for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 10.118828419885936478438... and c = 0.4250308979334609908... - Vaclav Kotesovec, Mar 19 2023

A361553 Expansion of g.f. A(x) satisfying 3*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

Original entry on oeis.org

1, 3, 24, 171, 1335, 11940, 115773, 1160901, 11901537, 124726644, 1332688035, 14455451526, 158660036535, 1758835084221, 19667067522966, 221573079684087, 2512635069594897, 28656903391830291, 328500210705228867, 3782806859877522522, 43738575934977450465
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 24*x^2 + 171*x^3 + 1335*x^4 + 11940*x^5 + 115773*x^6 + 1160901*x^7 + 11901537*x^8 + 124726644*x^9 + ...
where A = A(x) satisfies the doubly infinite sum
3*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,
3*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) * ...

Links

Crossrefs

Cf. A361550, A359920, A361552, A361554, A361555.

Programs

  • Mathematica
    (* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{r^3*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)) == k*r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + (2*s)/(-r + s^2) - 2/(s - r*s^3) + (-QPolyGamma[0, -Log[r*s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s^2]/Log[r^2], r^2] + QPolyGamma[0, Log[s^2/r]/Log[r^2], r^2]) / (s*Log[r]) == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(3*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), ", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(3*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), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 3*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) 3*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) 3*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) * 3^k for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 12.47776743014414138089586... and c = 0.474320402676760199022... - Vaclav Kotesovec, Mar 29 2023

A361554 Expansion of g.f. A(x) satisfying 4*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

Original entry on oeis.org

1, 4, 36, 296, 2732, 28980, 329996, 3872908, 46575260, 573472248, 7197096168, 91640952360, 1180636398320, 15364364313588, 201691201775092, 2667523242203932, 35510152549696208, 475424653523498396, 6397601663340197268, 86481499341290372804, 1173813146742741571560
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 36*x^2 + 296*x^3 + 2732*x^4 + 28980*x^5 + 329996*x^6 + 3872908*x^7 + 46575260*x^8 + 573472248*x^9 + ...
where A = A(x) satisfies the doubly infinite sum
4*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,
4*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) * ...

Links

Crossrefs

Cf. A361550, A359920, A361552, A361553, A361555.

Programs

  • PARI
    /* 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), ", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(4*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), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 4*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) 4*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) 4*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) * 4^k for n >= 0.

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)).

Original entry on oeis.org

1, 5, 50, 465, 4925, 59870, 776155, 10364135, 142082065, 1995371980, 28549274995, 414327073520, 6084353526535, 90258375062245, 1350607531232830, 20361436162127965, 308964002231172075, 4715119823819824535, 72324133311820587435, 1114404268419043050750
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

Examples

			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) * ...

Links

Crossrefs

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

Programs

  • Mathematica
    (* 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 *)
  • PARI
    /* 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), ", "))
  • PARI
    /* 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), ", "))

Formula

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)

A360191 G.f. 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2.

Original entry on oeis.org

1, 5, 18, 55, 149, 371, 867, 1923, 4086, 8374, 16634, 32152, 60669, 112041, 202943, 361200, 632647, 1091917, 1859225, 3126242, 5195715, 8541624, 13899866, 22404091, 35787815, 56683294, 89061028, 138872410, 214984454, 330532633, 504869316, 766357010, 1156355165
Offset: 0

Views

Author

Paul D. Hanna, Jan 29 2023

Keywords

Comments

Self-convolution inverse of A080332.

Examples

			G.f.: A(x) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + 16634*x^10 + 32152*x^11 + 60669*x^12 + ...

Links

Crossrefs

Cf. A080332, A361535, A361050, A361550.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[1/Product[(1 - x^k)^3 * (1 - x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
nmax = 30; CoefficientList[Series[1/(QPochhammer[x] * EllipticTheta[4, 0, x]^2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
  • PARI
    {a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^3 * (1 - x^(2*m-1))^2 +x*O(x^n)), n)}
for(n=0,32,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(!) A(x) = 1 / [Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2].
(2) A(x) = 1 / [Sum_{n=-oo..+oo} (6*n + 1) * x^(n*(3*n + 1)/2)].
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Feb 07 2023

A361535 Expansion of g.f. 1 / Product_{n>=1} ((1 - x^n)^6 * (1 - x^(2*n-1))^4).

Original entry on oeis.org

1, 10, 61, 290, 1172, 4212, 13833, 42262, 121625, 332764, 871641, 2197936, 5359005, 12679730, 29200593, 65617892, 144189054, 310400110, 655669910, 1360910666, 2779007594, 5589070978, 11081585154, 21679798590, 41883282555, 79958881544, 150943109191, 281926365224
Offset: 0

Views

Author

Paul D. Hanna, Mar 18 2023

Keywords

Examples

			G.f.: A(x) = 1 + 10*x + 61*x^2 + 290*x^3 + 1172*x^4 + 4212*x^5 + 13833*x^6 + 42262*x^7 + 121625*x^8 + 332764*x^9 + 871641*x^10 + ...
A related series begins
A(x)^(1/2) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + ... + A360191(n)*x^n + ...

Links

Crossrefs

Cf. A360191, A361050, A361550.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/((1 - x^k)^6 * (1 - x^(2*k-1))^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 19 2023 *)
  • PARI
    {a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^6 * (1 - x^(2*m-1))^4 + x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ exp(4*Pi*sqrt(n/3)) / (2^(5/2) * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Mar 19 2023
Showing 1-8 of 8 results.

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