A096374 Number of partitions of n such that the least part occurs with even multiplicity.

0, 1, 0, 3, 1, 5, 4, 11, 8, 19, 19, 35, 36, 59, 65, 104, 115, 168, 196, 276, 321, 440, 521, 694, 821, 1072, 1277, 1644, 1957, 2477, 2959, 3705, 4411, 5472, 6516, 8014, 9524, 11620, 13789, 16724, 19798, 23860, 28202, 33815, 39864, 47579, 55979, 66520, 78080

Offset

1,4

Comments

Also number of partitions of n such that the difference between the two largest distinct parts is even (it is assumed that 0 is a part in each partition). Example: a(6)=5 because we have [6],[5,1],[4,2],[2,2,2] and [3,1,1,1]. - Emeric Deutsch, Apr 04 2006

Links

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

Formula

G.f.: Sum_{m>=1} ((x^(2*m)/(1+x^m))/Product_{i>=m}(1-x^i)).

a(n) + A096375(n) = A000041(n).

Example

a(6) = 5 because we have [4,1,1], [3,3], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1].

Maple

g:=sum(x^(2*k)/(1+x^k)/product(1-x^j, j=k..70), k=1..50): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=1..48); # Emeric Deutsch, Apr 04 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i<1, 0, `if`(irem(n, i, 'r')=0
      and irem(r, 2)=0, 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[ EvenQ[ 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[n == 0 || i < 1, 0, {q, r} = QuotientRemainder[n, i]; If[r == 0 && Mod[q, 2] == 0, 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 *)
Table[Count[IntegerPartitions[n], _?(EvenQ[Length[Split[#][[-1]]]]&)], {n, 50}] (* Harvey P. Dale, Jun 02 2019 *)

Prog

(PARI) {q=sum(m=1, 100, (x^(2*m)/(1+x^m))/prod(i=m, 100, 1-x^i, 1+O(x^60)), 1+O(x^60)); for(n=1, 48, print1(polcoeff(q, n), ", "))} \\ Klaus Brockhaus, Jul 21 2004

Crossrefs

Cf. A000041, A096375.

Sequence in context: A082129 A363233 A159970 * A007085 A094648 A096975

Adjacent sequences: A096371 A096372 A096373 * A096375 A096376 A096377

Keyword

easy,nonn

Author

Vladeta Jovovic, Jul 19 2004

Extensions

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 21 2004

Status

approved