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%I #20 Dec 28 2015 05:34:43
%S 1,1,3,2,6,6,11,11,22,23,37,42,65,76,111,127,182,217,294,351,471,562,
%T 734,881,1137,1364,1733,2074,2608,3127,3883,4644,5732,6838,8367,9963,
%U 12113,14395,17396,20614,24785,29314,35059,41360,49270,57979,68775,80753
%N Number of partitions of n such that the least part occurs with odd multiplicity.
%H Alois P. Heinz, <a href="/A096375/b096375.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{m>=1} ((x^m/(1+x^m))/Product_{i>=m}(1-x^i)).
%F A096374(n) + a(n) = A000041(n).
%p b:= proc(n, i) option remember; `if`(i<1, 0, `if`(irem(n, i, 'r')=0
%p and irem(r, 2)=1, 1, 0)+ add(b(n-i*j, i-1), j=0..n/i))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Feb 27 2013
%t 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 *)
%t 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_ *)
%o (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
%Y Cf. A000041, A096374.
%K easy,nonn
%O 1,3
%A _Vladeta Jovovic_, Jul 19 2004
%E Edited and extended by _Robert G. Wilson v_ and _Klaus Brockhaus_, Jul 23 2004
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