A080054 -id:A080054 - cp's OEIS frontend

A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2k+1))/(1 - x^(2k+1)))^n.

1, 2, 8, 44, 256, 1512, 9056, 54896, 335872, 2069774, 12827888, 79875996, 499305472, 3131436856, 19694403520, 124165133424, 784478240768, 4965659813668, 31484486937512, 199923173603596, 1271192603065856, 8092551782518688, 51574780342740256, 329022223268286288, 2100934234342260736

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == 2*p + 2 (mod p^3) holds for all primes p >= 5. Cf. A270919. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Main diagonal of A289522.

Cf. A080054, A008705, A270919, A361008, A362408.

Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^n, {k, 0, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^n, {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == QPochhammer[-r*s, r^2*s^2] / QPochhammer[r*s, r^2*s^2], QPochhammer[r*s, r^2*s^2] + QPochhammer[r*s, r^2*s^2]*((QPolyGamma[0, Log[-r*s]/Log[r^2*s^2], r^2*s^2] - QPolyGamma[0, Log[r*s]/Log[r^2*s^2], r^2*s^2]) / Log[r^2*s^2]) + 2*r^2*s^2*Derivative[0, 1][QPochhammer][r*s, r^2*s^2] == 2*r^2*s*Derivative[0, 1][QPochhammer][-r*s, r^2*s^2]}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)

a(n) = A289522(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 6.52085730573545526010335599231748172235904... and c = 0.296494808714349908707366708893... - Vaclav Kotesovec, Aug 30 2017

A292037 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 2, 2, 6, 10, 16, 30, 46, 78, 124, 196, 306, 470, 724, 1086, 1644, 2438, 3608, 5304, 7734, 11232, 16196, 23270, 33206, 47250, 66846, 94232, 132280, 184966, 257720, 357768, 495090, 682702, 938760, 1286668, 1758708, 2397012, 3258340, 4417570, 5974204, 8059824

Offset: 0

Vaclav Kotesovec, Sep 08 2017

nonn

Convolution of A263140 and A035528 (with a(0)=1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A035528, A080054, A263140, A292038.

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/24 - Pi^4/(1344*Zeta(3)) + Pi^2 * n^(1/3) / (8*(7*Zeta(3))^(1/3)) + 3*(7*Zeta(3))^(1/3) * n^(2/3)/4) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(5/4) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962.

A103261 Number of partitions of 2n into parts with 10 types c^1 c^2...C^10 of each part. The even parts appear with multiplicity 1 for each type . The odd parts occur with multiplicity 2 for each part.

Original entry on oeis.org

1, 20, 200, 1360, 7200, 32024, 125280, 443680, 1450240, 4435940, 12827888, 35346800, 93377920, 237675640, 585229760, 1398704736, 3253934080, 7386124520, 16392493800, 35634450320, 75992326592, 159199081600, 328027789600

Offset: 0

Noureddine Chair, Feb 16 2005

nonn

This is also Sequence(A080054)^(10) or sequence(A007096)^(5).

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

			a(2)=200 because we have 10 types of 4, 45 ways of writing 4 in terms of ten of 2's only or ten of 11's only and 100 ways of writing 2's combined with 11's so the total number of ways of writing 4 is 200.

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A261648 (j=5), A014970 (j=6), A014972 (j=8).

series(product(((1+x^k)*(1-x^(2*k)))^(10)/((1-x^k)*(1+x^(2*k)))^(10),k=1..100),x=0,100);

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^10,{k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Aug 28 2015 *)

G.f.:(theta_4(0, x^2)/theta_4(0, x))^10= (theta_3(0, x)/theta_4(0, x))^5.

a(n) ~ exp(Pi*sqrt(5*n)) * 5^(1/4) / (64 * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

Example corrected by Vaclav Kotesovec, Sep 01 2015

A210030 Expansion of phi(-q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, -2, 4, 6, -8, -12, 16, 22, -30, -40, 52, 68, -88, -112, 144, 182, -228, -286, 356, 440, -544, -668, 816, 996, -1210, -1464, 1768, 2128, -2552, -3056, 3648, 4342, -5160, -6116, 7232, 8538, -10056, -11820, 13872, 16248, -18996, -22176, 25844, 30068

Offset: 0

Michael Somos, Mar 16 2012

sign

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

			1 - 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 8*q^5 - 12*q^6 + 16*q^7 + 22*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080015, A080054, A208850.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 17 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5, n))}

Expansion of eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5 in powers of q.
Euler transform of period 8 sequence [ -2, -3, -2, 2, -2, -3, -2, 0, ...].
G.f.: (Sum_k (-1)^k * x^k^2) / (Sum_k x^(2 * k^2)).
a(n) = (-1)^n * A080015(n) = (-1)^[(n + 1) / 4] * A080054(n).
Convolution inverse of A208850.

A261647 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^3.

Original entry on oeis.org

1, 6, 18, 44, 102, 216, 428, 816, 1494, 2650, 4584, 7740, 12804, 20808, 33264, 52400, 81462, 125100, 189966, 285516, 425016, 627040, 917436, 1331856, 1919332, 2746926, 3905784, 5519352, 7754064, 10833192, 15055216, 20817600, 28647414, 39241336, 53517060

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

Cf. A015128, A156616.
Cf. A080054, A007096, A014969, A261648, A014970, A014972, A103261.
Cf. A261610, A261649, A261651.
Cf. A261611, A261650, A261652.

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(3*n/2)) * 3^(1/4) / (8 * 2^(1/4) * n^(3/4)).

A261648 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^5.

Original entry on oeis.org

1, 10, 50, 180, 550, 1512, 3820, 9040, 20310, 43670, 90472, 181540, 354180, 674040, 1254640, 2289104, 4101430, 7228020, 12546030, 21473940, 36281656, 60565920, 99974140, 163297520, 264110180, 423211938, 672244600, 1059013320, 1655274320, 2568068120

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)).

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A014970 (j=6), A014972 (j=8), A103261 (j=10).

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^5,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(5*n/2)) * 5^(1/4) / (16 * 2^(1/4) * n^(3/4)).

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).

Original entry on oeis.org

1, 2, 2, 8, 14, 24, 52, 84, 158, 274, 464, 800, 1316, 2208, 3576, 5832, 9358, 14876, 23614, 36936, 57752, 89336, 137716, 210844, 321148, 486890, 733912, 1102336, 1646736, 2451464, 3632832, 5363988, 7889710, 11562712, 16888748, 24581904, 35670242, 51591096

Offset: 0

Vaclav Kotesovec, Sep 08 2017

nonn

Convolution of A262736 and A262811.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A080054, A262736, A262811, A284628, A285069, A292038.

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3*(7*Zeta(3))^(1/3)*n^(2/3) / 2^(5/3) - 1/12) * A * (7*Zeta(3))^(5/36) / (2^(31/36) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: exp(2*Sum_{k>=1} sigma_2(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596, 421844, 513834

Offset: 0

Noureddine Chair, Dec 20 2004; revised Jan 05 2005

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).
There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos, Oct 27 2008

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Cristina Ballantine, Mircea Merca, Jacobi's Four and Eight Squares Theorems and Partitions into Distinct Parts, Mediterranean Journal of Mathematics (2019) Vol. 16, No. 2, 26.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Ramanujan Theta Functions

Cf. A015128, A098151, A080054.

series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))),k=1..100),x=0,100);

nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(2/(QPochhammer[x] QPochhammer[-1, -x]) + O[x]^45)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 10 2005 */

Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos, Feb 10 2005
G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).
Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Oct 27 2008
Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Oct 27 2008
a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Aug 30 2015
G.f.: 2/((x; x)inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016
Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - Michael Somos, Nov 22 2016

A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369, 36189

Offset: 0

Michael Somos, Mar 02 2012

nonn

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

Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.

			1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...
a(5) = 6 since  5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
a(6) = 8 since  5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A115671, A187154, A208851.

A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n,0,50}] (* G. C. Greubel, Jun 19 2017 *)

{a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.
Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].
a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).

A215594 Expansion of f(-x, -x^4) / f(x, x^4) in powers of x where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, -2, 2, -2, 0, 2, -4, 6, -4, 0, 6, -12, 14, -10, 0, 14, -26, 30, -22, 0, 28, -52, 60, -42, 0, 54, -100, 112, -78, 0, 100, -180, 202, -140, 0, 174, -314, 350, -240, 0, 296, -532, 588, -402, 0, 492, -876, 966, -658, 0, 794, -1412, 1550, -1050, 0, 1260, -2232

Offset: 0

Michael Somos, Aug 16 2012

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

Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054 and A098151. - Peter Bala, Jun 10 2025

			1 - 2*x + 2*x^2 - 2*x^3 + 2*x^5 - 4*x^6 + 6*x^7 - 4*x^8 + 6*x^10 - 12*x^11 + ...
From _Peter Bala_, Jun 10 2025: (Start)
G.f.: A(q) = f(-q, -q^4) / f(q, q^4).
Simple continued fraction expansions of A(1/(2*m)):
  m = 2 [0; 1   1  2    7  1  1   1  10   1  2   12     82  1  6  1   48 ...]
  m = 3 [0; 1   2  2   17  1  1   2  14   2  2   24    318  1  1  1    1 ...]
  m = 4 [0; 1   3  2   31  1  1   3  18   3  2   40    810  2  7  2  161 ...]
  m = 5 [0; 1   4  2   49  1  1   4  22   4  2   60   1654  2  1  1    1 ...]
  m = 6 [0; 1   5  2   71  1  1   5  26   5  2   84   2946  3  7  1    1 ...]
  m = 7 [0; 1   6  2   97  1  1   6  30   6  2  112   4782  3  1  1    1 ...]
  m = 8 [0; 1   7  2  127  1  1   7  34   7  2  144   7258  4  7  1    2 ...]
  m = 9 [0; 1   8  2  161  1  1   8  38   8  2  180  10470  4  1  1    1 ...]
 m = 10 [0; 1   9  2  199  1  1   9  42   9  2  220  14514  5  7  1    3 ...]
 m = 11 [0; 1  10  2  241  1  1  10  46  10  2  264  19486  5  1  1    1 ...]
 m = 12 [0; 1  11  2  287  1  1  11  50  11  2  312  25482  6  7  1    4 ...]
 ...
The sequence of the 4th partial denominators [7, 17, 31, 49, ...] appears to be given by the polynomial 2*m^2 - 1 for m >= 2.
The sequence of the 11th partial denominators [12, 24, 40, 60, ...] appears to be given by the polynomial 2*(m^2 + m) for m >= 2.
The sequence of the 12th partial denominators [82, 318, 810, 1654, ...] appears to be given by the polynomial 2*(8*m^3 - 8*m^2 + 6*m - 3) for m >= 2.
The sequence of the 16th partial denominators appears to become quasi-polynomial in m for m >= 5. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A098151.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A215594[n_] := SeriesCoefficient[f[-x, -x^4]/f[x, x^4], {x, 0, n}]; Table[A215594[n], {n,0,50}] (* G. C. Greubel, Jun 18 2017 *)

{a(n) = local(A, s); if( n<0, 0, A = x * O(x^n); s = sqrtint( 40*n + 9); polcoeff( sum( k=(-s + 6)\10, (s - 3)\10, (-1)^k *  x^((5*k + 3)*k/2), A) / sum( k=(-s + 6)\10, (s - 3)\10,  x^((5*k + 3)*k/2), A), n))}

Euler transform of period 10 sequence [ -2, 1, 0, -2, 0, -2, 0, 1, -2, 0, ...].

a(5*n + 4) = 0.

cp's OEIS Frontend

A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A292037 Expansion of Product_{k>=1} ((1 + x^(2*k-1)) / (1 - x^(2*k-1)))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A103261 Number of partitions of 2n into parts with 10 types c^1 c^2...C^10 of each part. The even parts appear with multiplicity 1 for each type . The odd parts occur with multiplicity 2 for each part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A210030 Expansion of phi(-q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A261647 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A261648 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A292038 Expansion of Product_{k>=1} ((1 + x^(2*k-1)) / (1 - x^(2*k-1)))^(2*k-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

Views

Author

Keywords

Comments

Examples

A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2k+1))/(1 - x^(2k+1)))^n.

A292037 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^k.

A261647 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^3.

A261648 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^5.

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).