A027998 -id:A027998 - cp's OEIS frontend

A380291 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^2) is the g.f. of A027998.

1, 1, 9, 64, 425, 3026, 21672, 157095, 1149289, 8464240, 62683134, 466307865, 3482008904, 26083955002, 195932407939, 1475267031164, 11131100990825, 84140066313620, 637054366975740, 4830417047590165, 36674477204674750, 278779034863684377, 2121418004609211361, 16159262748227985561

Offset: 0

Peter Bala, Jan 19 2025

Given an integer sequence {f(n) : n >= 0} with f(0) = 1, there is a unique power series F(x) with rational coefficients, where F(0) = 1, such that f(n) = [x^n] F(x)^n. F(x) is given by F(x) = series_reversion(x/E(x)), where E(x) = exp(Sum_{n >= 1} f(n)*x^n/n). Furthermore, if the series E(x) has integer coefficients then the series F(x) also has integer coefficients and the sequence {f(n)} satisfies the Gauss congruences: f(n*p^r) == f(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r (by Stanley, Ch. 5, Ex. 5.2(a), p. 72 and the Lagrange inversion formula).
Thus the present sequence satisfies the Gauss congruences. In fact, stronger congruences appear to hold for the present sequence.
We conjecture that a(p) == 1 (mod p^3) for all primes p >= 7 (checked up to p = 61).
More generally, we conjecture that the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 7 and positive integers n and r. Some examples are given below.

			Examples of supercongruences:
a(7) - a(1) = 157095 - 1 = 2*(7^3)*229 == 0 (mod 7^3)
a(11) - a(1) = 466307865 - 1 = (2^3)*(11^3)*43793 == 0 (mod 11^3)
a(3*7) - a(3) = 278779034863684377 - 64 = (7^4)*43*26891*100413601 == 0 (mod 7^4)

R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

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

Cf. A027998, A078307, A380290.

Cf. A270913, A270922.

with(numtheory):
s_3 := n-> add((-1)^(n/d+1)*d^3, d in divisors(n)):
G(x) := series(exp(add(s_3(k)*x^k/k, k = 1..23)), x, 24):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..23);

Table[SeriesCoefficient[Product[(1 + x^k)^(n*k^2), {k, 1, n}], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
(* or *)
Table[SeriesCoefficient[Exp[n*Sum[Sum[(-1)^(k/d + 1)*d^3, {d, Divisors[k]}]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = [x^n] exp(n*Sum_{k >= 1} s_3(k)*x^k/k), where s_3(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3 = A078307(n).

a(n) ~ c * d^n / sqrt(n), where d = 7.7846790125019502578773343468308844201627754275100035492213697757399421948... and c = 0.2484592487737716543953469621097743519172686743284742545545347906986158... - Vaclav Kotesovec, Jul 30 2025

A285224 Indices of primes in A027998.

Original entry on oeis.org

3, 4, 5, 7, 11, 28, 45, 47, 71, 115, 135, 205, 230, 258, 408, 409, 665, 703, 915, 934, 3912, 5031, 5746, 6078, 6391, 7979, 11267, 11296, 14666, 17555, 22073, 23583, 30029, 36150, 51253, 54365, 57557, 68475

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

No other terms below 100000.

			11 is in the sequence because A027998(11) = 13309 is prime.

Cf. A027998, A035359.

A304049 Numbers k such that A027998(k) is divisible by k.

Original entry on oeis.org

1, 2, 14, 75, 292, 421, 1841, 3101, 4466, 60605, 87403

Offset: 1

Vaclav Kotesovec, May 05 2018

No other terms below 100000.

			75 is in the sequence because A027998(75) = 1005156449936335004175 = 13402085999151133389 * 75.

Cf. A027998.

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699

Offset: 0

N. J. A. Sloane

In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016

			For n = 4, we have 8 partitions
  01: [4]
  02: [4']
  03: [4'']
  04: [4''']
  05: [3, 1]
  06: [3', 1]
  07: [3'', 1]
  08: [2, 2']

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
Vaclav Kotesovec, Graph - The asymptotic ratio
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. 18.

Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Cf. A027998, A248882, A248883, A248884.
Cf. A026011, A027346, A027906.
Column k=1 of A284992.

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Apr 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

N=66; q='q+O('q^N);
gf= prod(n=1,N, (1+q^n)^n );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
G.f.: Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006
a(n) ~ zeta(3)^(1/6) * exp((3/2)^(4/3) * zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

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

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Original entry on oeis.org

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (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. 22.

Cf. A023872, A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^3),{k,1,nmax}],{x,0,nmax}],x]

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018
Euler transform of A309335. - Georg Fischer, Nov 10 2020

A206622 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^2).

Original entry on oeis.org

1, 2, 10, 36, 118, 376, 1148, 3376, 9654, 26894, 73192, 195188, 510948, 1315048, 3332720, 8326448, 20529526, 49998884, 120379574, 286726340, 676057144, 1578880480, 3654180236, 8385122192, 19085029540, 43103203626, 96630606968, 215105226728, 475608824400

Offset: 0

Paul D. Hanna, Feb 10 2012

nonn

Compare g.f. to: Product_{n>0} (1+x^n)/(1-x^n) = exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ) which equals 1/theta_4(x) = 1/(1 + 2*Sum_{n>=1} (-x)^(n^2)).
Convolution of A023871 and A027998. - Vaclav Kotesovec, Aug 19 2015
In general, if g.f. = Product_{k>=1} ((1 + x^k)/(1 - x^k))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(Pi * 2^(5/4) * c2^(1/4) * n^(3/4) / 3 + 7*c1 * Zeta(3) * sqrt(n) / (Pi^2 * sqrt(2*c2)) + (c0*Pi / (2^(5/4) * c2^(1/4)) - 49*c1^2 * Zeta(3)^2 / (2^(5/4) * c2^(5/4) * Pi^5)) * n^(1/4) + 22411 * c1^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - 7*c0*c1 * Zeta(3) / (4*c2 * Pi^2) - c2 * Zeta(3) / (4*Pi^2) + c1/12) * Pi^(c1/12) * c2^(1/8 + c0/8 + c1/48) / (A^c1 * 2^(15/8 + 11*c0/8 + 7*c1/48) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 5. Cf. A380582. - Peter Bala, Jan 21 2025

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 36*x^3 + 118*x^4 + 376*x^5 + 1148*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^4/(1-x^2)^4 * (1+x^3)^9/(1-x^3)^9 *...
Also, A(x) = Euler transform of [2,7,18,28,50,63,98,112,162,175,...]:
A(x) = 1/((1-x)^2*(1-x^2)^7*(1-x^3)^18*(1-x^4)^28*(1-x^5)^50*(1-x^6)^63*...).

Seiichi Manyama, Table of n, a(n) for n = 0..9042 (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. A156616, A206623, A206624, A001158 (sigma_3), A380582.

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

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 3)-sigma(m, 3))/4*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+7*x+12*x^2+7*x^3+2*x^4)/(1-x^2+x*O(x^n))^3);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,35,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_3(2*n) - sigma_3(n))/4 * x^n/n ), where sigma_3(n) is the sum of cubes of divisors of n (A001158).
The inverse Euler transform has g.f.: x*(2 + 7*x + 12*x^2 + 7*x^3 + 2*x^4)/(1-x^2)^3.
a(n) ~ exp(2^(5/4)*Pi*n^(3/4)/3 - Zeta(3)/(4*Pi^2)) / (2^(15/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A007331(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

Original entry on oeis.org

1, -1, -3, -6, 0, 11, 42, 63, 73, -45, -267, -720, -1095, -1239, -66, 2794, 8757, 16017, 22885, 19634, -2359, -61979, -161867, -302190, -421971, -432051, -126712, 690578, 2278273, 4584989, 7269985, 8965464, 7515373, -845659, -19930400, -53474765, -100195759

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n^2, g(n) = -1. - Seiichi Manyama, Nov 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A027998, A281590, A281591.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), this sequence (m=2), A284897 (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^2) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^2))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078307(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

G.f.: exp(Sum_{k>=1} (-1)^k*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 30 2018

A078307 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.

Original entry on oeis.org

1, 7, 28, 55, 126, 196, 344, 439, 757, 882, 1332, 1540, 2198, 2408, 3528, 3511, 4914, 5299, 6860, 6930, 9632, 9324, 12168, 12292, 15751, 15386, 20440, 18920, 24390, 24696, 29792, 28087, 37296, 34398, 43344, 41635, 50654, 48020, 61544, 55314, 68922, 67424

Offset: 1

Vladeta Jovovic, Nov 22 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Index entries for sequences mentioned by Glaisher.

Cf. A000593, A078306, A027998.

with(numtheory):
a:= n-> add((-1)^(n/d+1)*d^3, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Aug 03 2013

a[n_] := Sum[(-1)^(n/d+1)*d^3, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jan 17 2014 *)
f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := (6*8^e + 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 42] (* Amiram Eldar, Oct 27 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^3); \\ Indranil Ghosh, Apr 05 2017

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**3 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{n >= 1} n^3*x^n/(1+x^n).
Multiplicative with a(2^e) = (6*8^e+1)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4/2880 = 0.236758... . - Amiram Eldar, Oct 27 2022

A027999 Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf.

Original entry on oeis.org

1, 0, 1, 3, 6, 13, 24, 49, 91, 181, 334, 632, 1163, 2138, 3880, 7006, 12531, 22279, 39369, 69078, 120597, 209282, 361405, 620829, 1061687, 1807014, 3062642, 5168784, 8688820, 14549659, 24274226, 40353748, 66854518, 110391391, 181695436, 298129605, 487706902

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A027998, A028377, A258349, A258341, A258344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(binomial(i, 2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[Binomial[i, 2], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(-2025 * Zeta(3)^3 / (98*Pi^8) - 135*(15/7)^(1/4) * Zeta(3)^2 / (28*Pi^5) * n^(1/4) - 3*sqrt(15/7) * Zeta(3) / (2*Pi^2) * sqrt(n) + 2*(7/15)^(1/4) * Pi/3 * n^(3/4)), where Zeta(3) = A002117. - Vaclav Kotesovec, May 27 2015

cp's OEIS Frontend

A380291 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^2) is the g.f. of A027998.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A285224 Indices of primes in A027998.

Views

Author

Keywords

Comments

Examples

Crossrefs

A304049 Numbers k such that A027998(k) is divisible by k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A206622 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

Views