A077285 Number of partitions of n with designated summands.

1, 1, 3, 5, 10, 15, 28, 41, 69, 102, 160, 231, 352, 498, 732, 1027, 1470, 2031, 2856, 3896, 5382, 7272, 9896, 13233, 17800, 23579, 31362, 41219, 54288, 70791, 92456, 119698, 155097, 199512, 256664, 328134, 419436, 533162, 677412, 856573, 1082284, 1361679

Offset

0,3

Comments

Sum of products of multiplicities of parts in all partitions of n. The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 1,1,2,2,4 and their sum is a(4) = 10. - Vladeta Jovovic, Feb 16 2005

Links

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

G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.

William Y. C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y. Y. Shen, On the Number of Partitions with Designated Summands, arXiv:1208.2210 [math.CO], 2012.

Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3, arXiv:2101.04058 [math.CO], 2021. See also Integers (2023) Vol. 23, Art. No. A43.

N. J. A. Sloane, Transforms

Formula

Expansion of eta(q^6) / (eta(q) * eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Feb 05 2004

Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 2, ...]. - Michael Somos, Feb 05 2004

G.f.: P(x)*P(x^2)*P(x^3)/P(x^6), where P(x)=Product_{k>0} 1/(1-x^k) is the partition generating function (A000041).

Equals EULER(DCONV(A000012, iEULER(A000027))).

G.f.: Product_{i>=1} (1-x^i+x^(2*i)) / (1-x^i)^2. - Vladeta Jovovic, Jan 16 2005

a(n) ~ 5^(3/4) * exp(Pi*sqrt(10*n)/3) / (2^(11/4) * 3^(3/2) * n^(5/4)). - Vaclav Kotesovec, Nov 28 2015

a(n) = Sum_{k>=1} k*A266477(n,k). - Alois P. Heinz, Dec 29 2015

G.f.: Product_{i>0} (1 + Sum_{j>0} j*x^(j*i)). - Seiichi Manyama, Oct 08 2017

Example

a(3)=5 because the partitions of 3 with designated summands are 3', 2'1', 1'11, 11'1, 111'.
1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 28*x^6 + 41*x^7 + 69*x^8 + 102*x^9 + ...

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

Mathematica

max = 50; f = Product[(1-x^i+x^(2*i))/(1-x^i)^2, {i, 1, max}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014, after Vladeta Jovovic *)
nmax=100; CoefficientList[Series[Product[(1+x^(3*k)) / ((1-x^k) * (1-x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
QP = QPochhammer; s = QP[q^6]/(QP[q]*QP[q^2]*QP[q^3]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
Table[Total[l = Tally /@ IntegerPartitions@n;
Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 41}] (* Robert Price, Jun 06 2020 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))} /* Michael Somos, Feb 05 2004 */

Crossrefs

Cf. A000041, A091601.

Cf. A102186 (partitions into odd parts with designated summands).

Cf. A258210, A266477.

Sequence in context: A326472 A326597 A008337 * A072523 A054473 A265508

Adjacent sequences: A077282 A077283 A077284 * A077286 A077287 A077288

Keyword

nonn

Author

Jorn B. Olsson (olsson(AT)math.ku.dk), Nov 26 2003

Extensions

Edited and extended by Christian G. Bower, Jan 23 2004

Status

approved