A102186 The PDO(n) function (Partitions with Designated summands in which all parts are Odd): the sum of products of multiplicities of parts in all partitions of n into odd parts.

1, 1, 2, 4, 5, 8, 12, 16, 22, 32, 42, 56, 76, 98, 128, 168, 213, 272, 348, 436, 548, 688, 852, 1056, 1308, 1603, 1964, 2404, 2920, 3544, 4296, 5176, 6230, 7488, 8958, 10704, 12772, 15182, 18024, 21368, 25254, 29808, 35136, 41308, 48504, 56880, 66552, 77776

Offset: 0

Author

Vladeta Jovovic, Feb 16 2005

Keywords

• easy
• nonn

Examples

			a(8)=22 because in the six partitions of 8 into odd parts, namely, 71,53,5111,3311,311111,11111111, the multiplicities of the parts are (1,1),(1,1),(1,3),(2,2),(1,5),(8) with products 1,1,3,4,5,8, having sum 22.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.
• Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 10, 12.
• Nayandeep Deka Baruah and Kanan Kumari Ojah, Partitions with designated summands in which all parts are odd, INTEGERS 15 (2015), #A9.
• Shishuo Fu and James Sellers, A refined view of a curious identity for partitions into odd parts with designated summands, arXiv:2505.21111 [math.CO], 2025.

Crossrefs

Cf. A077285 (partitions with designated summands).

Programs

Maple

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

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[b[n - i*j, i - 2]*j, {j, 1, n/i}]]]; a[n_] := b[n, Quotient[1 + n, 2]*2 - 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1+x^(3*k)) / ((1-x^k) * (1+x^(6*k))), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
Table[Total[l = Tally /@ Select[IntegerPartitions@n, VectorQ[#, OddQ] &];
  Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 47}] (* Robert Price, Jun 08 2020 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)*eta(x^6+A)^2/ eta(x+A)/eta(x^3+A)/eta(x^12+A), n))} /* Michael Somos, Jul 30 2006 */

Formula

Euler transform of period 12 sequence [1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, ...].

a(n) ~ 5^(1/4) * exp(sqrt(5*n)*Pi/3) / (2^(5/2)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 28 2015

G.f.: Product_{k>=1} (1 + Sum_{j>=1} j * x^(j*(2*k - 1))). - Ilya Gutkovskiy, Nov 06 2019

Extensions

More terms from Emeric Deutsch, Mar 28 2005

Name expanded by N. J. A. Sloane, Nov 21 2015