A288418 -id:A288418 - cp's OEIS frontend

A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

1, 1, 3, 7, 14, 28, 58, 106, 201, 372, 669, 1187, 2101, 3624, 6229, 10591, 17796, 29659, 49107, 80492, 131157, 212237, 341084, 544883, 865717, 1367233, 2148552, 3359490, 5227270, 8096544, 12486800, 19174319, 29326306, 44678825, 67811375, 102549673, 154545549

Offset: 0

Joerg Arndt, Jun 24 2011

nonn

Euler transform of A002131. - Vaclav Kotesovec, Mar 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
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.

Cf. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, Jul 03 2017 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

N=66;  x='x+O('x^N);
Q(x)=prod(k=1,N,1+x^k);
gf=prod(k=1,N, Q(x^k)^k );
Vec(gf) /* Joerg Arndt, Jun 24 2011 */

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 09 2017

a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A288417 a(n) = Sum_{d|n} A000593(n/d).

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 9, 4, 18, 14, 13, 15, 15, 18, 35, 5, 19, 36, 21, 21, 45, 26, 25, 20, 38, 30, 58, 27, 31, 70, 33, 6, 65, 38, 63, 54, 39, 42, 75, 28, 43, 90, 45, 39, 126, 50, 49, 25, 66, 76, 95, 45, 55, 116, 91, 36, 105, 62, 61, 105, 63, 66, 162, 7, 105, 130, 69

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000012 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000012.

Sum_{d|n} d^k*A000593(n/d): this sequence (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

f[p_, e_] := Sum[(i + 1)*p^(e - i), {i, 0, e}]; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

a(n)={sumdiv(n, d, sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

L.g.f.: log(Product_{k>=1} (1 + x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
Multiplicative with a(2^e) = e+1 and a(p^e) = Sum_{i=0..e} (i+1)*p^(e-i) for e >= 0 and prime p > 2. - Werner Schulte, Jan 05 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/144 = 0.676452... . - Amiram Eldar, Oct 22 2022

A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).

Original entry on oeis.org

1, 3, 7, 7, 11, 21, 15, 15, 34, 33, 23, 49, 27, 45, 77, 31, 35, 102, 39, 77, 105, 69, 47, 105, 86, 81, 142, 105, 59, 231, 63, 63, 161, 105, 165, 238, 75, 117, 189, 165, 83, 315, 87, 161, 374, 141, 95, 217, 162, 258, 245, 189, 107, 426, 253, 225, 273, 177, 119, 539, 123, 189, 510, 127, 297

Offset: 1

Paul D. Hanna, Jun 26 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A060640, A107742.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), this sequence (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2]&]&]; Array[a, 65] (* Jean-François Alcover, Dec 23 2015 *)
f[p_, e_] := ((p + e*(p-1) - 2)*p^(e+1) + 1)/(p-1)^2; f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI
```
a(n)=sumdiv(n,d,d*sumdiv(d,m,m%2))
```

N=66; x='x+O('x^N); /* that many terms */
c=sum(j=1, N, j*x^j);
t=log( 1/prod(j=0, N, eta(x^(2*j+1))) );
gf=serconvol(t, c);
Vec(gf) /* show terms */
/* Joerg Arndt, May 03 2008 */

a(n) = Sum_{d|n} d * Sum_{m|d} (m mod 2).
G.f.: Sum_{n>=1} a(n)/n*x^n = Sum_{j>=1} Sum_{i>=1} log(1+x^(i*j)).
From Vladeta Jovovic, Jul 05 2005:(Start)
Multiplicative with a(2^e) = 2^(e+1)-1 and a(p^e) = (p^(e+2)*(e+1)-p^(e+1)*(e+2)+1)/(p-1)^2 for p>2.
G.f.: Sum_{n>0} n*A000005(n)*x^n/(1+x^n).
G.f.: Sum_{n>0} n*A001227(n)*x^n/(1-x^n).
a(n) = A060640(n) if n is odd, else a(n) = A060640(n) - 2*A060640(n/2).
a(n) = Sum_{d|n} d*A001227(d).
a(n) = Sum_{d|n} d*A000593(n/d).
A107742(n) = (1/n)*Sum_{k=1..n} a(k)*A107742(n-k). (End)

A288419 a(n) = Sum_{d|n} d^3*A000593(n/d).

Original entry on oeis.org

1, 9, 31, 73, 131, 279, 351, 585, 850, 1179, 1343, 2263, 2211, 3159, 4061, 4681, 4931, 7650, 6879, 9563, 10881, 12087, 12191, 18135, 16406, 19899, 22990, 25623, 24419, 36549, 29823, 37449, 41633, 44379, 45981, 62050, 50691, 61911, 68541, 76635, 68963, 97929

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000578 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000578, A027847, A183700, A288414.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), this sequence (k=3), A288420 (k=4).

f[p_, e_] := (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n)={sumdiv(n, d, (n/d)^3*sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027847(n) for odd n.
Multiplicative with a(2^e) = (8^(e+1)-1)/7 and a(p^e) = (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4*zeta(3)/2880 = (7/32)*zeta(3)*zeta(4) = (7/32) * A183700 = 0.284596... . (End)

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

Original entry on oeis.org

1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000583, A013662, A013663, A027848, A288415.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

Original entry on oeis.org

1, -1, -2, -2, 1, 5, 4, 10, 6, -5, -20, -27, -37, -32, -18, 23, 82, 128, 190, 185, 143, 43, -160, -424, -662, -968, -1058, -971, -571, 238, 1326, 2748, 4195, 5301, 5930, 5473, 3353, 55, -5346, -12106, -19421, -26603, -31950, -33248, -29344, -17469, 2343, 30966

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

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

Cf. A192065, A288418.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), this sequence (m=1), A288422 (m=2), A288423 (m=3).

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(1,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

nmax = 50; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k))) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A192065.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

cp's OEIS Frontend

A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

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A288417 a(n) = Sum_{d|n} A000593(n/d).

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A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)*x^n = log( Sum_{n>=0} A107742(n)*x^n ).

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A288419 a(n) = Sum_{d|n} d^3*A000593(n/d).

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A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

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A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

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A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).