A182819 -id:A182819 - cp's OEIS frontend

A182818 G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).

1, 3, 8, 19, 41, 83, 161, 299, 538, 942, 1610, 2694, 4427, 7153, 11387, 17884, 27741, 42543, 64565, 97034, 144519, 213432, 312720, 454803, 656835, 942364, 1343596, 1904354, 2684008, 3762667, 5248002, 7284132, 10063319, 13841107, 18956002

Offset: 0

Paul D. Hanna, Dec 05 2010

sigma(2n) = A000203(2n), the sum of divisors of 2n (A062731).
Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).
Number of partitions of n where there are 2 kinds of even parts and 3 kinds of odd parts. - Ilya Gutkovskiy, Jan 17 2018

			G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...
log(A(x)) = 3*x + 7*x^2/2 + 12*x^3/3 + 15*x^4/4 + 18*x^5/5 + 28*x^6/6 + 24*x^7/7 + 31*x^8/8 + ... + sigma(2n)*x^n/n + ...

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

Cf. A062731, A000203, A000041; variants: A182819, A182820, A182821.

with(combinat):
seq(add((-1)^k*numbpart(k)*numbpart(2*n - k), k = 0..2*n), n = 0..40);

nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)
nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^2, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

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

x='x+O('x^66); Vec(eta(x^2)/eta(x)^3) \\ Joerg Arndt, Dec 05 2010

G.f.: A(x) = E(x^2)/E(x)^3 where E(x)=Product_{n>=1} (1 - x^n). - Joerg Arndt, Dec 05 2010
Conjecture: exp( Sum_{n>=1} sigma(s*n)*x^n/n ) == Product_{d|s} eta(x^d)^(-moebius(d)*sigma(s/d)). - Joerg Arndt, Dec 05 2010
The ordinary generating function A(x) is the infinite product F(x) * F(x^2) * F(x^3) * ..., where F(x) is the ordinary generating function of A005408. - Gary W. Adamson, Jul 15 2012
a(n) ~ 5^(3/4) * exp(Pi*sqrt(5*n/3)) / (16 * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Nov 29 2015
From Peter Bala, Jan 24 2016: (Start)
a(n) = Sum_{k = 0..2*n} (-1)^k*p(k)*p(2*n-k), where p(n) = A000041(n) is the partition function.
A(x^2) = 1/Product_{n>=1} (1 - (-x)^n) * 1/Product_{n>=1} (1 - x^n). (End)
G.f.: A(x) = Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 follows directly from the above formula by Joerg Arndt. - Paul D. Hanna, Dec 07 2018

A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5n)x^n/n ).

Original entry on oeis.org

1, 6, 27, 98, 315, 917, 2486, 6345, 15427, 35965, 80897, 176296, 373652, 772381, 1561130, 3091476, 6008896, 11480887, 21591830, 40016045, 73157052, 132052382, 235535752, 415433365, 725043875, 1252857043, 2144601961, 3638413830

Offset: 0

Paul D. Hanna, Dec 05 2010

sigma(5*n) = A000203(5*n), the sum of divisors of 5n.

Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

			G.f.: A(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + ...
log(A(x)) = 6*x + 18*x^2/2 + 24*x^3/3 + 42*x^4/4 + 31*x^5/5 + 72*x^6/6 + 48*x^7/7 + 90*x^8/8 + ... + sigma(5n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000203, A000041; variants: A182818, A182819, A182820.
Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A277212 (k=5),
this sequence (k=6).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(5*j))/(1 - x^j)^6: j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

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

default(seriesprecision,66); Vec(eta(x^5)/eta(x)^6) \\ Joerg Arndt, Dec 05 2010

m=30; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^(5*j))/(1 - x^j)^6)) \\ G. C. Greubel, Nov 18 2018

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
s = prod((1 - x^(5*j))/(1 - x^j)^6 for j in (1..32))
list(s) # G. C. Greubel, Nov 18 2018

G.f.: A(x) = E(x^5)/E(x)^6 where E(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Dec 05 2010
a(n) ~ 29^(3/2) * exp(sqrt(58*n/15)*Pi) / (2400*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 28 2016
A(x^5) = P(x)*P(a*x)*P(a^2*x)*P(a^3*x)*P(a^4*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n), and where a = exp(2*Pi*i/5) is a primitive fifth root of unity. - Peter Bala, Jan 24 2017

A182820 G.f.: exp( Sum_{n>=1} sigma(4n)*x^n/n ).

Original entry on oeis.org

1, 7, 32, 119, 385, 1127, 3057, 7799, 18914, 43950, 98434, 213486, 450051, 925013, 1858355, 3657052, 7062245, 13404195, 25038741, 46086250, 83669927, 149970936, 265608168, 465149039, 806022315, 1382822644, 2350101516, 3958427938

Offset: 0

Paul D. Hanna, Dec 05 2010

sigma(4n) = A000203(4n), the sum of divisors of 4n.

Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

			G.f.: A(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 +...
log(A(x)) = 7*x + 15*x^2/2 + 28*x^3/3 + 31*x^4/4 + 42*x^5/5 + 60*x^6/6 + 56*x^7/7 + 63*x^8/8 +...+ sigma(4n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000203, A000041; variants: A182818, A182819, A182821.

with(combinat):
b := n -> add((-1)^k*numbpart(k)*numbpart(n-k), k = 0..n):
seq( add(I^k*b(k)*b(4*n-k), k = 0..4*n), n = 0..30 );
# - Peter Bala, Jan 24 2017

nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

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

x='x+O('x^66); Vec(eta(x^2)^3/eta(x)^7) \\ Joerg Arndt, Dec 05 2010

Generating function A(x) = E(x^2)^3/E(x)^7 where E(x) = Product_{n>=1} (1-x^n). [Joerg Arndt, Dec 05 2010]
a(n) ~ 11^(5/4) * exp(sqrt(11*n/3)*Pi) / (128 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Nov 28 2016
From Peter Bala, Jan 24 2016: (Start)
A(x^4) = P(x)*P(-x)*P(i*x)*P(-i*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n).
a(n) = Sum_{k = 0..4*n} i^k*b(k)*b(4*n-k), where b(n) := Sum_{k = 0..n} (-1)^k*p(k)*p(n-k). (End)

A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

Original entry on oeis.org

1, -4, 2, 9, -9, -2, 0, -5, 9, 9, 0, -9, -1, -9, 0, -1, 9, 9, -9, 9, 0, 9, -5, -18, -18, 9, 7, 0, 9, 0, 0, 9, 9, -18, 18, -7, -9, -9, -9, 9, -4, -9, -9, 18, 9, 0, 18, 9, 0, -9, -9, -8, -9, 18, -9, 9, -18, 1, -9, -18, 9, 0, 18, 18, 0, 0, 9, -9, 18, -9, 5, -9, 0, -9, -9, -9, -18, 11, 9

Offset: 0

Paul D. Hanna, Feb 16 2011

sign

			G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A182819, A144613, A000203.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 12 2018 *)

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

{a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}

G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by Joerg Arndt in A182819]
a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - Michael Somos, Jul 12 2018

A283119 Expansion of exp( Sum_{n>=1} sigma(6n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 12, 86, 469, 2141, 8594, 31247, 104945, 330094, 982284, 2786861, 7584060, 19893185, 50494558, 124437410, 298555264, 699017259, 1600364304, 3589048673, 7896510620, 17067607791, 36283650153, 75947406513, 156672628539, 318804641925, 640390347979

Offset: 0

Seiichi Manyama, Mar 01 2017

nonn

sigma(6*n) = A000203(6*n), the sum of divisors of 6*n (A224613).

			G.f.: A(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 + ...
log(A(x)) = 12*x + 28*x^2/2 + 39*x^3/3 + 60*x^4/4 + 72*x^5/5 + 91*x^6/6 + 96*x^7/7 + 124*x^8/8 + ... + sigma(6*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A224613 (sigma(6*n)), A283164 (exp( Sum_{n>=1} -sigma(6*n)*x^n/n )).

Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), this sequence (k=6), A283077 (k=7), A283120 (k=8), A283121 (k=9).

Table[SeriesCoefficient[Product[(1 - x^(2 i))^4*(1 - x^(3 i))^3/((1 - x^i)^12*(1 - x^(6 i))), {i, n}], {x, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Mar 01 2017 *)

G.f.: Product_{n>=1} (1 - x^(2*n))^4 * (1 - x^(3*n))^3/((1 - x^n)^12 * (1 - x^(6*n))).
a(n) = (1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 55^(7/4) * exp(sqrt(55*n)*Pi/3) / (41472*sqrt(3)*n^(9/4)). - Vaclav Kotesovec, Mar 20 2017

A283120 Expansion of exp( Sum_{n>=1} sigma(8n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 15, 128, 815, 4289, 19663, 81057, 306799, 1081986, 3594142, 11338690, 34193246, 99080387, 277046893, 750192227, 1973050940, 5053026949, 12628736331, 30859262181, 73849589786, 173333118663, 399528823032, 905418038792, 2019454523623, 4437187104779

Offset: 0

Seiichi Manyama, Mar 01 2017

nonn

			G.f.: A(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 + ...
log(A(x)) = 15*x + 31*x^2/2 + 60*x^3/3 + 63*x^4/4 + 90*x^5/5 + 124*x^6/6 + 120*x^7/7 + 127*x^8/8 + ... + sigma(8*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A283122 (sigma(8*n)), A283168 (exp( Sum_{n>=1} -sigma(8*n)*x^n/n )).

Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), A283077 (k=7), this sequence (k=8), A283121 (k=9).

G.f.: Product_{n>=1} (1 - x^(2*n))^7/(1 - x^n)^15.
a(n) = (1/n)*Sum_{k=1..n} sigma(8*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 529 * 23^(1/4) * exp(sqrt(23*n/3)*Pi) / (73728 * 3^(1/4) * n^(11/4)). - Vaclav Kotesovec, Mar 20 2017

A283121 Expansion of exp( Sum_{n>=1} sigma(9n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 13, 104, 633, 3224, 14404, 58151, 216294, 751582, 2464860, 7689669, 22961822, 65955677, 182985947, 492016590, 1285829996, 3274100475, 8139933477, 19795490575, 47165634583, 110259083454, 253208634687, 571880965638, 1271549402110, 2785836824325, 6019078365425

Offset: 0

Seiichi Manyama, Mar 01 2017

nonn

			G.f.: A(x) = 1 + 13*x + 104*x^2 + 633*x^3 + 3224*x^4 + 14404*x^5 + ...
log(A(x)) = 13*x + 39*x^2/2 + 40*x^3/3 + 91*x^4/4 + 78*x^5/5 + 120*x^6/6 + 104*x^7/7 + 195*x^8/8 + ... + sigma(9*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A283123 (sigma(9*n)), A283169 (exp( Sum_{n>=1} -sigma(9*n)*x^n/n )).

Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), A283077 (k=7), A283120 (k=8), this sequence (k=9).

G.f.: Product_{n>=1} (1 - x^(3*n))^4/(1 - x^n)^13.
a(n) = (1/n)*Sum_{k=1..n} sigma(9*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 1225 * sqrt(35) * exp(sqrt(70*n)*Pi/3) / (559872*n^3). - Vaclav Kotesovec, Mar 20 2017

A203321 L.g.f.: Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} x^n/n exp( Sum_{k>=1} sigma(nk)x^(n*k)/k ).

Original entry on oeis.org

1, 3, 7, 19, 26, 75, 78, 211, 241, 518, 463, 1447, 1002, 2558, 2612, 5715, 3928, 11901, 7316, 21574, 17031, 35159, 23047, 80575, 40951, 108488, 86911, 206638, 107823, 370220, 173725, 570803, 372181, 816496, 451883, 1723741, 665150, 2048982, 1404150, 3705366, 1530859, 5892479

Offset: 1

Paul D. Hanna, Dec 31 2011

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 75*x^6/6 +...
where
L(x) = x*exp(1*x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 +...) +
x^2/2*exp(3*x^2 + 7*x^4/2 + 12*x^6/3 + 15*x^8/4 + 18*x^10/5 +...) +
x^3/3*exp(4*x^3 + 12*x^6/2 + 13*x^9/3 + 28*x^12/4 + 24*x^15/5 +...) +
x^4/4*exp(7*x^4 + 15*x^8/2 + 28*x^12/3 + 31*x^16/4 + 42*x^20/5 +...) +
x^5/5*exp(6*x^5 + 18*x^10/2 + 24*x^15/3 + 42*x^20/4 + 31*x^25/5 +...) +
x^6/6*exp(12*x^6 + 28*x^12/2 + 39*x^18/3 + 60*x^24/4 + 72*x^30/5 +...) +
x^7/7*exp(8*x^7 + 24*x^14/2 + 32*x^21/3 + 56*x^28/4 + 48*x^35/5 +...) +
x^8/8*exp(15*x^8 + 31*x^16/2 + 60*x^24/3 + 63*x^32/4 + 90*x^40/5 +...) +...
...
Equivalently, L(x) = Sum_{n>=1} P_n(x^n) * x^n/n where
P_n(x) = exp( Sum_{k>=1} sigma(n*k)*x^k/k ), which begin:
P_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...;
P_2(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...;
P_3(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 + 533*x^6 +...;
P_4(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 + 3057*x^6 +...;
P_5(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + 2486*x^6 +...;
P_6(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 +...;
P_7(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 +...;
P_8(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 +...;
...

Paul D. Hanna, Table of n, a(n) for n = 1..120

Cf. A203320; A000041, A182818, A182819, A182820, A182821.

{a(n)=local(L=vector(max(n,1), i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), sigma(m*k)*x^(m*k)/k)+x*O(x^n))))); if(n<1,0,L[n])}

{a(n)=local(A=1+x+x*O(x^n),P=exp(sum(k=1,n,sigma(k)*x^k/k)+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(P, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * P_n(x^n) where P_n(x^n) = Product_{k=0..n-1} P(u^k*x) where u is an n-th root of unity, and P(x) is the partition function (A000041); P(x) = exp(Sum_{n>=1} sigma(n)*x^n/n) where sigma(n) is the sum of divisors of n (A000203).

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22527, 62329, 164516, 416948, 1019690, 2416246, 5565864, 12498215, 27421815, 58903768, 124088548, 256749822, 522450250, 1046735092, 2066948472, 4026431543, 7743987036, 14715788745, 27648250012, 51390298666, 94550761844

Offset: 0

Seiichi Manyama, Feb 28 2017

nonn

			G.f.: A(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + ...
log(A(x)) = 8*x + 24*x^2/2 + 32*x^3/3 + 56*x^4/4 + 48*x^5/5 + 96*x^6/6 + 57*x^7/7 + 120*x^8/8 + ... + sigma(7*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282942 (Product_{n>=1} (1 - x^n)^8/(1 - x^(7*n))), A283078 (sigma(7*n)).

Cf. exp( Sum_{n>=1} sigma(k*n)*x^n/n ): A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), this sequence (k=7), A283120 (k=8), A283121 (k=9).

G.f.: exp( Sum_{n>=1} sigma(7*n)*x^n/n ).
a(n) = (1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 3025 * exp(sqrt(110*n/21)*Pi) / (28224*sqrt(14)*n^(5/2)). - Vaclav Kotesovec, Mar 20 2017

A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(nk) x^(n*k)/k ) ).

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 35, 61, 124, 222, 427, 749, 1434, 2493, 4585, 8032, 14511, 25096, 44791, 77019, 135435, 232002, 402957, 685582, 1181399, 1998168, 3410288, 5741978, 9726821, 16286497, 27409625, 45672026, 76378731, 126706567, 210690588, 347954716, 575685559, 946756712

Offset: 0

Paul D. Hanna, Dec 31 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 35*x^6 + 61*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} P_n(x^n) * x^n/n )
where P_n(x) = exp( Sum_{k>=1} sigma(n*k)*x^k/k ), which begin:
P_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...;
P_2(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...;
P_3(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 + 533*x^6 +...;
P_4(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 + 3057*x^6 +...;
P_5(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + 2486*x^6 +...;
P_6(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 +...;
P_7(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 +...;
P_8(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 +...;
...
Also, P_n(x^n) = Product_{k=0..n-1} P(u^k*x) where u = n-th root of unity:
P_1(x) = P(x), the partition function;
P_2(x^2) = P(x)*P(-x);
P_3(x^3) = P(x)*P(u*x)*P(u^2*x) where u = exp(2*Pi*I/3);
P_4(x^4) = P(x)*P(I*x)*P(I^2*x)*P(I^3*x) where I^2 = -1;
...
The logarithmic derivative of this sequence begins:
A203321 = [1,3,7,19,26,75,78,211,241,518,463,1447,1002,2558,...].

Paul D. Hanna, Table of n, a(n) for n = 0..120

Cf. A203321; A000041, A182818, A182819, A182820, A182821.

{a(n)=local(L=vector(n+1, i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), sigma(m*k)*x^(m*k)/k)+x*O(x^n))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)}

{a(n)=local(A=1+x+x*O(x^n),P=exp(sum(k=1,n,sigma(k)*x^k/k)+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(P, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}

G.f.: exp( Sum_{n>=1} P_n(x^n) * x^n/n ) where P_n(x^n) = Product_{k=0..n-1} P(u^k*x), u is an n-th root of unity, and P(x) is the partition function (A000041); P(x) = exp(Sum_{n>=1} sigma(n)*x^n/n) where sigma(n) is the sum of divisors of n (A000203).

The logarithmic derivative yields A203321.

cp's OEIS Frontend

A182818 G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).

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A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5*n)*x^n/n ).

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A182820 G.f.: exp( Sum_{n>=1} sigma(4n)*x^n/n ).

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A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

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A283119 Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x.

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A283120 Expansion of exp( Sum_{n>=1} sigma(8*n)*x^n/n ) in powers of x.

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A283121 Expansion of exp( Sum_{n>=1} sigma(9*n)*x^n/n ) in powers of x.

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A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5n)x^n/n ).

A283119 Expansion of exp( Sum_{n>=1} sigma(6n)x^n/n ) in powers of x.

A283120 Expansion of exp( Sum_{n>=1} sigma(8n)x^n/n ) in powers of x.

A283121 Expansion of exp( Sum_{n>=1} sigma(9n)x^n/n ) in powers of x.

A203321 L.g.f.: Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} x^n/n exp( Sum_{k>=1} sigma(nk)x^(n*k)/k ).

A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(nk) x^(n*k)/k ) ).