A015128 -id:A015128 - cp's OEIS frontend

A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

1, 2, 34, 228, 1414, 8872, 52876, 301136, 1662614, 8929406, 46738920, 239036116, 1197187780, 5882369976, 28397283056, 134864166352, 630819797174, 2908948327780, 13236421303742, 59477002686404, 264104800719672, 1159649708139680, 5037895127964316

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023873 and A248883. - Vaclav Kotesovec, Aug 19 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
If m is even and m >= 2, then can be simplified as: a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + (-1)^(m/2) * Gamma(m+1) * Zeta(m+1) / (2^(m+1) * Pi^m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^16/(1-x^2)^16 * (1+x^3)^81/(1-x^3)^81 *...
Also, A(x) = Euler transform of [2,31,162,496,1250,2511,4802,7936,...]:
A(x) = 1/((1-x)^2*(1-x^2)^31*(1-x^3)^162*(1-x^4)^496*(1-x^5)^1250*(1-x^6)^2511*...).

Seiichi Manyama, Table of n, a(n) for n = 0..2916 (terms 0..1000 from Vaclav Kotesovec)
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. 23.

Cf. A015128 (m=0), A156616 (m=1), A206622 (m=2), A206623 (m=3), A001160 (sigma_5).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^4)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 5)-sigma(m, 5))/16*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+31*x+152*x^2+341*x^3+460*x^4+341*x^5+152*x^6+31*x^7+2*x^8)/(1-x^2+x*O(x^n))^5); polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_5(2*n) - sigma_5(n))/16 * x^n/n ), where sigma_5(n) is the sum of 5th powers of divisors of n (A001160).
Inverse Euler transform has g.f.: x*(2 + 31*x + 152*x^2 + 341*x^3 + 460*x^4 + 341*x^5 + 152*x^6 + 31*x^7 + 2*x^8)/(1-x^2)^5.
a(n) ~ exp(3*2^(2/3)*Pi*n^(5/6)/5 + 3*Zeta(5)/(4*Pi^4)) / (2^(7/6) * 3^(1/2) * n^(7/12)), where Zeta(5) = A013663. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A207641 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).

Original entry on oeis.org

1, 1, 3, 5, 9, 15, 25, 39, 61, 93, 139, 205, 299, 429, 611, 861, 1201, 1663, 2285, 3115, 4221, 5683, 7605, 10123, 13405, 17661, 23163, 30245, 39323, 50925, 65699, 84445, 108167, 138089, 175719, 222921, 281965, 355627, 447309, 561139, 702133, 876395, 1091301

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

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

In Ramanujan's equation let a = x and b = 1. - Michael Somos, Nov 20 2015

			G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 39*x^7 +...
such that, by definition,
A(x) = 1 + x*(1+x)/(1-x) + x^2*(1+x)*(1+x^2)/((1-x)*(1-x^2)) + x^3*(1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) +...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..750 from Vaclav Kotesovec)
Frank Garvan, Dyson's rank function and Andrews's SPT-function, slides 29, 30.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A015128, A059777.

a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
a[ n_] := SeriesCoefficient[ 1 / ((1 + x) EllipticTheta[ 4, 0, x]), {x, 0, n}]; (* Michael Somos, Nov 20 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^m*prod(k=1,m,(1+x^k)/(1-x^k +x*O(x^n))) ),n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / ((1 + x) * eta(x + A)^2), n))}; /* Michael Somos, Nov 20 2015 */

Expansion of 1 / ((1 + x) * phi(-x)) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Nov 20 2015
G.f.: 1 + x*(1+x) * (1 / (1-x)^2 + 2*x^3 / ((1-x)*(1-x^2))^2 + 2*x^7*(1+x) / ((1-x)*(1-x^2)*(1-x^3))^2 + 2*x^12*(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2 + ...). [Ramanujan] - Michael Somos, Nov 20 2015
a(n) + a(n+1) = A015128(n+1) for n >= 0. - Seiichi Manyama, Jul 12 2018
a(n) ~ exp(Pi*sqrt(n)) / (16*n). - Vaclav Kotesovec, Jun 18 2019

A235793 Sum of all parts of all overpartitions of n.

Original entry on oeis.org

2, 8, 24, 56, 120, 240, 448, 800, 1386, 2320, 3784, 6048, 9464, 14560, 22080, 32992, 48688, 71064, 102600, 146720, 207984, 292336, 407744, 564672, 776650, 1061424, 1442016, 1947904, 2617192, 3498720, 4654464, 6163584, 8126448, 10669472, 13952400, 18175896

Offset: 1

Omar E. Pol, Jan 18 2014

nonn

The equivalent sequence for partitions is A066186.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A015128, A066186, A235790, A235792, A235797, A235798, A236000.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0$2], b(n, i-1)+add((l-> l+[0, l[1]*i*j])
       (2*b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 21 2014

Table[n*Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 1, 40}] (* Jean-François Alcover, Oct 20 2016, after Vaclav Kotesovec *)

a(n) = n*A015128(n).

a(n) ~ exp(Pi*sqrt(n)) / 8. - Vaclav Kotesovec, May 19 2018

A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

Original entry on oeis.org

2, 4, 2, 10, 4, 2, 20, 8, 4, 2, 38, 16, 8, 4, 2, 68, 30, 16, 8, 4, 2, 118, 52, 28, 16, 8, 4, 2, 196, 88, 48, 28, 16, 8, 4, 2, 318, 144, 82, 48, 28, 16, 8, 4, 2, 504, 230, 132, 80, 48, 28, 16, 8, 4, 2, 782, 360, 208, 128, 80, 48, 28, 16, 8, 4, 2, 1192, 552, 324, 202, 128, 80, 48, 28, 16, 8, 4, 2

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that row n lists the first differences of row n of triangle A235797 together with 2 (as the final term of the row).

The equivalent sequence for partitions is A066633.

			Triangle begins:
2;
4,   2;
10,  4,  2;
20,  8,  4,  2;
38, 16,  8,  4,  2;
68, 30, 16,  8,  4,  2;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums give A235792.

Cf. A015128, A066633, A235790, A235793, A235797, A236000, A236001.

A(n)={my(p=prod(k=1, n, (1 + x^k)/(1 - x^k) + O(x*x^n))); Mat(vector(n, k, Col(2*(p + O(x*x^(n-k)))*x^k/((1 - x^k)*(1 + x^k)), -n)))}
{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

G.f. of column k: 2*(x^k/((1 - x^k)*(1 + x^k))) * Product_{j>0} (1 + x^j)/(1 - x^j). - Andrew Howroyd, Feb 19 2020

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

A261519 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2^k).

Original entry on oeis.org

1, 4, 16, 60, 208, 692, 2224, 6940, 21152, 63188, 185488, 536268, 1529648, 4310804, 12017264, 33171916, 90745472, 246201412, 662897232, 1772295020, 4707336848, 12426673188, 32617079280, 85152717404, 221183486496, 571784014244, 1471463190032, 3770577250716

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A034899 and A102866.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
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. 27.

Cf. A156616, A261520, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(2^(2*j)-1)) = 0.2545212486386431009939814261118792033...

A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 200, 428, 902, 1874, 3780, 7504, 14732, 28368, 54052, 101960, 189750, 349996, 640218, 1159624, 2084952, 3722008, 6593560, 11606268, 20308188, 35312170, 61065636, 105060200, 179795936, 306244136, 519291476, 876554860, 1473504846

Offset: 0

Vaclav Kotesovec, Dec 15 2015

nonn

Convolution of A022629 and A006906.

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

Cf. A006906, A015128, A022629, A022661, A022693, A261584.

nmax = 40; CoefficientList[Series[Product[(1 + k*x^k)/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(n/3), where
c = 28711548.45004804552683870974706458425598... if mod(n,3) = 0
c = 28711547.74098394497470795294574937283075... if mod(n,3) = 1
c = 28711547.58138731567204220029302329316039... if mod(n,3) = 2.

A306045 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k).

Original entry on oeis.org

1, 2, 10, 74, 682, 7562, 98410, 1463114, 24367402, 449039882, 9069093610, 199050295754, 4713774570922, 119735740542602, 3246094020405610, 93519923311825994, 2852458136048627242, 91805618091515859722, 3108657616523130770410, 110453876295411957125834

Offset: 0

Vaclav Kotesovec, Jun 18 2018

nonn

Convolution of A167137 and A305550.

Vaclav Kotesovec, Table of n, a(n) for n = 0..410

Cf. A015128, A167137, A305550, A306046.

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k) / (1 - (Exp[x] - 1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A015128(k) * k!.

a(n) ~ n! * exp(Pi^2 * (1 - log(2)) / (16*log(2)) + Pi * sqrt(n/(2*log(2)))) / (8*n*(log(2))^n).

A131942 Number of partitions of n in which each odd part has odd multiplicity.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 6, 11, 13, 21, 24, 35, 44, 59, 74, 99, 126, 158, 202, 250, 320, 392, 495, 598, 758, 908, 1134, 1358, 1685, 2003, 2466, 2925, 3576, 4234, 5129, 6064, 7308, 8612, 10305, 12135, 14443, 16963, 20085, 23548, 27754, 32482, 38105, 44503, 52042

Offset: 0

Brian Drake, Jul 30 2007

			a(5)=6 because 5, 4+1, 3+2, 2+2+1, 2+1+1+1 and 1+1+1+1+1 have all odd parts with odd multiplicity. The partition 3+1+1 is the partition of 5 which is not counted.

Brian Drake and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Cf. A000041, A015128, A006950, A046682.

A:= series(product( 1/(1-q^(2*n)) *(1+q^(2*n-1)-q^(4*n-2))/(1-q^(4*n-2)), n=1..15),q,25): seq(coeff(A,q,i), i=0..24);

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/ ((1-x^(2*k)) * (1-x^(4*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{n>=1} (1+q^(2n-1)-q^(4n-2))/((1-q^(2n))(1-q^(4n-2))).

a(n) ~ sqrt(Pi^2 + 8*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*n/2)) / (8*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

A230441 Number of overpartitions of n minus the number of partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 9, 17, 29, 49, 78, 124, 190, 288, 427, 627, 905, 1296, 1831, 2567, 3563, 4910, 6709, 9112, 12286, 16473, 21953, 29108, 38388, 50398, 65850, 85683, 111020, 143302, 184263, 236113, 301498, 383757, 486909, 615955, 776921, 977263, 1225934, 1533945

Offset: 0

Omar E. Pol, Jan 09 2014

nonn

Number of overpartitions of n that contain at least one overlined part. - Omar E. Pol, Jan 19 2014

			The 14 overpartitions of 4 are
01: [4],
02: [4'],
03: [2, 2],
04: [2', 2],
05: [3, 1],
06: [3', 1],
07: [3, 1'],
08: [3', 1'],
09: [2, 1, 1],
10: [2', 1, 1],
11: [2, 1', 1],
12: [2', 1', 1],
13: [1, 1, 1, 1],
14: [1', 1, 1, 1].
There are 9 overpartitions that contain at least one overlined part, so a(4) = 9. - _Omar E. Pol_, Jan 19 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A015128, A235792.

b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0$2],
      b(n, i-1) +add((l->l+[0, l[2]])(b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> (l->l[2]-l[1])(b(n$2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2014

b[n_, i_] := b[n, i] = If[n==0, {1, 1}, If[i<1, {0, 0}, b[n, i-1] + Sum[Function[ {l}, l+{0, l[[2]]}][b[n-i*j, i-1]], {j, 1, n/i}]]]; a[n_] := Function[{l}, l[[2]]-l[[1]]][b[n, n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 28 2015, after Alois P. Heinz *)

a(n) = A015128(n) - A000041(n).

A236000 Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 2, 2, 2, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 1

Omar E. Pol, Jan 18 2014

In the data section the overlined parts cannot be represented correctly, therefore the sequence represents all possible suborderings generated by the overlined parts.
The diagram in the second part of the Example section shows only one of the possible suborderings.
The equivalent sequence for partitions is A211992.
The equivalent sequence for compositions is A228525.
See both sequences for more information.
Row n contains A015128(n) overpartitions.
Row n contains A235792(n) parts.
Row sums give A235793.

			Triangle begins:
[1], [1];
[1, 1], [1, 1], [2], [2];
[1, 1, 1], [1, 1, 1], [2, 1], [2, 1], [2, 1], [2, 1], [3], [3];
[1, 1, 1, 1], [1, 1, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [3, 1], [3, 1], [3, 1], [3, 1], [2, 2], [2, 2], [4], [4];
...
Illustration of initial terms (n: 1..4)
-----------------------------------------
n      Diagram          Overpartition
-----------------------------------------
.       _
1      |.|              1',
1      |_|              1;
.       _ _
2      |.| |            1', 1,
2      |_| |            1,  1,
2      |  .|            2',
2      |_ _|            2;
.       _ _ _
3      |.| | |          1', 1,  1,
3      |_| | |          1,  1,  1,
3      |  .|.|          2', 1',
3      |   |.|          2,  1',
3      |  .| |          2', 1,
3      |_ _| |          2,  1,
3      |    .|          3',
3      |_ _ _|          3;
.       _ _ _ _
4      |.| | | |        1', 1,  1,  1,
4      |_| | | |        1,  1,  1,  1,
4      |  .|.| |        2', 1', 1,
4      |   |.| |        2,  1', 1,
4      |  .| | |        2', 1,  1,
4      |_ _| | |        2,  1,  1,
4      |    .|.|        3', 1',
4      |     |.|        3,  1',
4      |    .| |        3', 1,
4      |_ _ _| |        3,  1,
4      |  .|   |        2', 2,
4      |_ _|   |        2,  2,
4      |      .|        4',
4      |_ _ _ _|        4;
.

Cf. A015128, A211992, A228525, A235790, A235792, A235793, A235797, A235798, A236001

cp's OEIS Frontend

A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

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A207641 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).

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A235793 Sum of all parts of all overpartitions of n.

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A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

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A261519 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2^k).

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A265758 Expansion of Product_{k>=1} ((1 + k*x^k)/(1 - k*x^k)).

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A306045 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k).

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A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).