A265844 -id:A265844 - cp's OEIS frontend

A077335 Sum of products of squares of parts in all partitions of n.

1, 1, 5, 14, 46, 107, 352, 789, 2314, 5596, 14734, 34572, 92715, 210638, 531342, 1250635, 3042596, 6973974, 16973478, 38399806, 91301956, 207992892, 483244305, 1089029008, 2533640066, 5642905974, 12912848789, 28893132440, 65342580250, 144803524640

Offset: 0

Vladeta Jovovic, Nov 30 2002

nonn

			The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products of squares of parts are 16,9,16,4,1 and their sum is a(4) = 46.

Seiichi Manyama, Table of n, a(n) for n = 0..3134 (terms 0..1000 from Alois P. Heinz)

Cf. A006906, A074141, A092484, A265844, A292164.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 07 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, i^2*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[Total[Times@@(#^2)&/@IntegerPartitions[n]],{n,0,30}] (* Harvey P. Dale, Apr 29 2018 *)
Table[Total[Times@@@(IntegerPartitions[n]^2)],{n,0,30}] (* Harvey P. Dale, Sep 07 2023 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

N=22;q='q+O('q^N); Vec(1/prod(n=1,N,1-n^2*q^n)) \\ Joerg Arndt, Aug 31 2015

G.f.: 1/Product_{m>0} (1 - m^2*x^m).
Recurrence: a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(2*k/d + 1).
a(n) = S(n,1), where S(n,m) = n^2 + Sum_{k=m..n/2} k^2*S(n-k,k), S(n,n) = n^2, S(n,m) = 0 for m > n. - Vladimir Kruchinin, Sep 07 2014
From Vaclav Kotesovec, Mar 16 2015: (Start)
a(n) ~ c * 3^(2*n/3), where
c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k
c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1
c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2
(End)
In closed form, a(n) ~ (Product_{k>=4}(1/(1 - k^2/3^(2*k/3))) / ((1 - 3^(-2/3)) * (1 - 4*3^(-4/3))) + Product_{k>=4}(1/(1 - (-1)^(2*k/3)*k^2/3^(2*k/3))) / ((-1)^(2*n/3) * (1 + 4/3*(-1/3)^(1/3)) * (1 - (-1/3)^(2/3))) + Product_{k>=4}(1/(1 - (-1)^(4*k/3)*k^2/3^(2*k/3))) / ((-1)^(4*n/3) * (1 + (-1)^(1/3)*3^(-2/3)) * (1 - 4*(-1)^(2/3)*3^(-4/3)))) * 3^(2*n/3 - 1). - Vaclav Kotesovec, Apr 25 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

Original entry on oeis.org

1, 1, 4, 13, 25, 77, 161, 393, 726, 2010, 3850, 7874, 16791, 31627, 69695, 139560, 255997, 482277, 986021, 1716430, 3544299, 6507128, 11887340, 21137849, 38636535, 70598032, 123697772, 233003286, 412142276, 711896765, 1252360770

Offset: 0

Jon Perry, Apr 04 2004

nonn

Sum of squares of products of terms in all partitions of n into distinct parts.

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding squares of products are 36, 25, 64, 36 and their sum is a(6) = 161.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A022629, A077335, A265844, A285737, A292165.

Column k=2 of A292189.

b:= proc(n, i) option remember; (m->
      `if`(mn, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* Robert G. Wilson v, Apr 05 2005 *)

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ Seiichi Manyama, Sep 10 2017

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

More terms from Robert G. Wilson v, Apr 05 2004

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).

Original entry on oeis.org

1, 2, 40, 1800, 149024, 21223800, 4609532520, 1414165715200, 581109518753920, 307788983933760954, 204081628466048180200, 165541724073121026987224, 161233041454793035411134240, 185663865439487951708529417080, 249499302292252719726304186789160

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

Convolution of A292304 and A292417.

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A265758, A265844, A292304, A292417, A292419.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^2*x^k)/(1-n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, ((1+n^2*x^k)/(1-n^2*x^k) +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ 2 * n^(2*n) * (1 + 2/n^2 + 4/n^4 + 8/n^6 + 14/n^8 + 24/n^10), for coefficients see A015128.

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A077335 Sum of products of squares of parts in all partitions of n.

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A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

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A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).

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cp's OEIS Frontend

A077335 Sum of products of squares of parts in all partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

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Author

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Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k) / (1 - n^2*x^k).

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Mathematica

PARI

Formula

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).