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
Keywords
Examples
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 + ...
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
Crossrefs
Programs
-
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 *) -
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 */
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).
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
Extensions
Edited and extended by Christian G. Bower, Jan 23 2004
Comments