A109386 -id:A109386 - cp's OEIS frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954

Offset: 0

Vladeta Jovovic, Jun 11 2005

From Gus Wiseman, Sep 13 2022: (Start)
Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,2}}        {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,2}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
Intervals are counted by A001227, ranked by A073485.
The initial version is A007294.
The strict version is A327731.
The version for gapless multisets instead of intervals is A356941.
The case of strict partitions is A356957.
Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}    {{3}}        {{4}}
         {{1,1}}  {{1,1,1}}    {{2,2}}
                  {{1},{2}}    {{1},{3}}
                  {{1},{1,1}}  {{1,1,1,1}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
Constant multisets are counted by A000005, ranked by A000961.
The non-strict version is A006171.
The unlabeled version is A089259.
The non-constant block version is A261049.
The version for twice-partitions is A279786, factorizations A296131.
Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,1,1}}      {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
The strict version is A327731 (also).
(End)

Vaclav Kotesovec, 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.
N. J. A. Sloane, Transforms

Cf. A006171, A109386, A219554, A280473, A280486, A288007.
Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A072233 counts partitions by sum and length.
Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)

a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */

N=66;  x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

Euler transform of A001227.
Weigh transform of A000005.
G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009
G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017
Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

More terms from Paul D. Hanna, Jun 26 2005

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

A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, 0, 2, 2, -1, -2, 0, -1, -1, -4, -1, 2, 0, -1, 2, 2, 5, -1, 4, 8, -4, -5, 0, -1, -1, -6, -1, 3, -7, -9, -5, 1, 3, -3, 3, 17, 0, -6, 8, 12, 8, 0, 8, 17, -11, -9, -10, 0, -2, -20, 5, 14, -18, -25, -10, 1, -7, -21, 2, 29, -12, -17, 6, 17, 32, -4

Offset: 0

Seiichi Manyama, Jun 04 2017

sign

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

Cf. A107742, A109386.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): this sequence (m=0), A288421 (m=1), A288422 (m=2), A288423 (m=3).

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[(&*[1 + x^(j*k): j in [1..2*m]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018

with(numtheory): seq(coeff(series(exp(-add(sigma(k)*x^k/(k*(1-x^(2*k))),k=1..n)),x,n+1), x, n), n = 0 .. 70); # Muniru A Asiru, Jan 30 2019

A109386[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2] &] &]; a[0] = 1; a[n_] := a[n] = -(1/n) Sum[A109386[k] a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 04 2017 *)
CoefficientList[Series[1/Product[Product[1+x^(j*k), {j,1,100}], {k,1,100}], {x,0,80}], x] (* G. C. Greubel, Oct 29 2018 *)

m=80; x='x+O('x^m); Vec(1/(prod(k=1,2*m, prod(j=1,2*m, 1+x^(j*k) )))) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A107742.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 26 2018

A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

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

Cf. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A001001(n) for odd n.
Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (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

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99

Offset: 1

Ilya Gutkovskiy, Sep 13 2019

Inverse Moebius transform of A002131.

Dirichlet convolution of A000027 with A001227.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]

a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} A002131(d).
a(n) = Sum_{d|n} d * A001227(n/d).
a(n) = (A007429(n) + A288417(n)) / 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022

cp's OEIS Frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

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

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A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

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A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

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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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A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

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