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%I #63 Jun 14 2023 17:30:19
%S 1,1,3,5,10,15,28,41,69,102,160,231,352,498,732,1027,1470,2031,2856,
%T 3896,5382,7272,9896,13233,17800,23579,31362,41219,54288,70791,92456,
%U 119698,155097,199512,256664,328134,419436,533162,677412,856573,1082284,1361679
%N Number of partitions of n with designated summands.
%C 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
%H Alois P. Heinz, <a href="/A077285/b077285.txt">Table of n, a(n) for n = 0..10000</a>
%H G. E. Andrews, R. P. Lewis, and J. Lovejoy, <a href="http://dx.doi.org/10.4064/aa105-1-5">Partitions with designated summands</a>, Acta Arith. 105 (2002), no. 1, 51-66.
%H William Y. C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y. Y. Shen, <a href="http://arxiv.org/abs/1208.2210">On the Number of Partitions with Designated Summands</a>, arXiv:1208.2210 [math.CO], 2012.
%H Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, <a href="https://arxiv.org/abs/2101.04058">Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3</a>, arXiv:2101.04058 [math.CO], 2021. See also <a href="http://math.colgate.edu/~integers/x43/x43.pdf">Integers</a> (2023) Vol. 23, Art. No. A43.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Expansion of eta(q^6) / (eta(q) * eta(q^2) * eta(q^3)) in powers of q. - _Michael Somos_, Feb 05 2004
%F Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 2, ...]. - _Michael Somos_, Feb 05 2004
%F 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).
%F Equals EULER(DCONV(A000012, iEULER(A000027))).
%F G.f.: Product_{i>=1} (1-x^i+x^(2*i)) / (1-x^i)^2. - _Vladeta Jovovic_, Jan 16 2005
%F 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
%F a(n) = Sum_{k>=1} k*A266477(n,k). - _Alois P. Heinz_, Dec 29 2015
%F G.f.: Product_{i>0} (1 + Sum_{j>0} j*x^(j*i)). - _Seiichi Manyama_, Oct 08 2017
%e a(3)=5 because the partitions of 3 with designated summands are 3', 2'1', 1'11, 11'1, 111'.
%e 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 + ...
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +add(b(n-i*j, i-1)*j, j=1..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Feb 26 2013
%t 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_ *)
%t 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 *)
%t 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 *)
%t Table[Total[l = Tally /@ IntegerPartitions@n;
%t Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 41}] (* _Robert Price_, Jun 06 2020 *)
%o (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 */
%Y Cf. A000041, A091601.
%Y Cf. A102186 (partitions into odd parts with designated summands).
%Y Cf. A258210, A266477.
%K nonn
%O 0,3
%A Jorn B. Olsson (olsson(AT)math.ku.dk), Nov 26 2003
%E Edited and extended by _Christian G. Bower_, Jan 23 2004
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