A096375 Number of partitions of n such that the least part occurs with odd multiplicity.

1, 1, 3, 2, 6, 6, 11, 11, 22, 23, 37, 42, 65, 76, 111, 127, 182, 217, 294, 351, 471, 562, 734, 881, 1137, 1364, 1733, 2074, 2608, 3127, 3883, 4644, 5732, 6838, 8367, 9963, 12113, 14395, 17396, 20614, 24785, 29314, 35059, 41360, 49270, 57979, 68775, 80753

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{m>=1} ((x^m/(1+x^m))/Product_{i>=m}(1-x^i)).

A096374(n) + a(n) = A000041(n).

Maple

b:= proc(n, i) option remember; `if`(i<1, 0, `if`(irem(n, i, 'r')=0
      and irem(r, 2)=1, 1, 0)+ add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013

Mathematica

f[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, If[ OddQ[ Count[ p[[k]], p[[k]][[ -1]] ]], c++ ]; k++ ]; c]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Jul 23 2004 *)
b[n_, i_] := b[n, i] = If[i<1, 0, {q, r} = QuotientRemainder[n, i]; If[r == 0 && Mod[q, 2] == 1, 1, 0] + Sum[b[n - i*j, i-1], {j, 0, n/i}]] ; a[n_] := b[n, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Prog

(PARI) {q=sum(m=1, 100, (x^m/(1+x^m))/prod(i=m, 100, 1-x^i, 1+O(x^60)), 1+O(x^60)); for(n=1, 47, print1(polcoeff(q, n), ", "))} - Klaus Brockhaus, Jul 21 2004

Crossrefs

Cf. A000041, A096374.

Sequence in context: A023360 A154028 A157793 * A062200 A114208 A014686

Adjacent sequences: A096372 A096373 A096374 * A096376 A096377 A096378

Keyword

easy,nonn

Author

Vladeta Jovovic, Jul 19 2004

Extensions

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 23 2004

Status

approved