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

0,3

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, J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.

Nayandeep Deka Baruah and Kanan Kumari Ojah, Partitions with designated summands in which all parts are odd, INTEGERS 15 (2015), #A9.

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

Example

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.

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 *)

Prog

(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 */

Crossrefs

Cf. A077285 (partitions with designated summands).

Sequence in context: A092268 A335702 A069259 * A039842 A188216 A238395

Adjacent sequences: A102183 A102184 A102185 * A102187 A102188 A102189

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 16 2005

Extensions

More terms from Emeric Deutsch, Mar 28 2005

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

Status

approved