A285928 -id:A285928 - cp's OEIS frontend

A093160 Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.

1, 4, 14, 40, 101, 236, 518, 1080, 2162, 4180, 7840, 14328, 25591, 44776, 76918, 129952, 216240, 354864, 574958, 920600, 1457946, 2285452, 3548550, 5460592, 8332425, 12614088, 18953310, 28276968, 41904208, 61702876, 90304598

Offset: 0

Michael Somos, Mar 26 2004, Apr 17 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 101*x^4 + 236*x^5 + 518*x^6 + 1080*x^7 + ...
G.f. = q + 4*q^3 + 14*q^5 + 40*q^7 + 101*q^9 + 236*q^11 + 518*q^13 + ...

A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 381, Section 488.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001938, A112143, A285927, A285928.

a[ n_] := SeriesCoefficient[ (Product[ 1 + x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}])^4, {x, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[m] / (4 Sqrt[1 - m]), {q, 0, n + 1/2}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ m^(1/4) / (2 (1 - Sqrt @ m)), {q, 0, n/2 + 1/4 }]];
s = (QPochhammer[q^4]/QPochhammer[q])^4 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = A * (1 + 16*A); A = 8*A2 + (1 + 32*A) * sqrt(A2)); polcoeff( sqrt(A/x), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^4, n))};

G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^4.
Expansion of q^(-1/2) * k / (4 * k') in powers of q where q is Jacobi's nome and k is the elliptic modulus.
Expansion of q^(-1/4) * k^(1/2) / (2 * (1 - k)) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.
Expansion of (psi(x^2) / phi(-x))^2 = (psi(x) / phi(-x^2))^4 = (psi(-x) / phi(-x))^4 = (psi(x^2) / psi(-x))^4 = (chi(x) / chi(-x^2)^2)^4 = ( chi(x) * chi(-x)^2)^-4 = (chi(-x) * chi(-x^2))^-4 = (f(-x^4) / f(-x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 4, 4, 4, 0, ...].
Given g.f. A(x), then B(x) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f. A(q) satisfies A(q) = sqrt(A(-q^2)) / (1 - 4*q*A(-q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. [Joerg Arndt, Aug 06 2011]
A001938(n) = (-1)^n * a(n). Convolution inverse of A112143.
a(n) ~ exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A046897(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 28 2017

A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 19, 42, 81, 155, 276, 486, 821, 1368, 2214, 3541, 5544, 8586, 13082, 19740, 29403, 43414, 63423, 91935, 132075, 188418, 266733, 375232, 524331, 728514, 1006216, 1382604, 1889739, 2570719, 3480420, 4691682, 6297102, 8418252, 11209347, 14870970

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), this sequence (m=3), A093160 (m=4), A285928 (m=5).

Cf. A000726, A199659.

nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k)) / (1 - x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.

Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) With[{k = 5}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

Original entry on oeis.org

1, -5, 5, 10, -15, -1, -30, 50, 65, -95, -1, -170, 220, 300, -380, 0, -635, 820, 1025, -1310, 0, -2045, 2525, 3140, -3845, 2, -5780, 7070, 8565, -10405, -1, -15130, 18125, 21760, -25960, 0, -36820, 43780, 51785, -61290, 0, -85170, 100030, 117500, -137550, 0

Offset: 0

Seiichi Manyama, Apr 29 2017

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 5*x + 5*x^2 + 10*x^3 - 15*x^4 - x^5 - 30*x^6 + 50*x^7 + 65*x^8 - 95*x^9 + ...
G.f. = q^-5 - 5*q + 5*q^7 + 10*q^13 - 15*q^19 - q^25 - 30*q^31 + 50*q^37 + 65*q^43 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

(Product_{k>0} (1 - x^k) / (1 - x^(m*k)))^m: A022597 (m=2), A199659 (m=3), A112143 (m=4), this sequence (m=5).

Cf. A285928.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^5]^5, {x, 0, n}]; (* Michael Somos, Apr 29 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^5, n))}; /* Michael Somos, Apr 29 2017 */

a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.
Expansion of q^(5/6) * eta(q)^5 / eta(q^5)^5 in powers of q. - Michael Somos, Apr 29 2017
Expansion of f(-x)^5 / f(-x^5)^5 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 29 2017
Euler transform of period 5 sequence [-5, -5, -5, -5, 0, ...]. - Michael Somos, Apr 29 2017

cp's OEIS Frontend

A093160 Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.

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A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

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A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

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A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

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A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

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