A218355 Number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is even.

1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 2, 6, 2, 8, 3, 9, 5, 12, 7, 13, 9, 16, 13, 19, 17, 22, 23, 25, 29, 30, 37, 35, 46, 41, 58, 49, 70, 57, 85, 68, 103, 81, 123, 97, 145, 115, 172, 139, 201, 164, 236, 197, 274, 234, 318, 280, 368, 330, 425, 394, 488, 463, 561, 548, 644, 642, 738, 755, 844, 879, 965, 1029

Offset: 0

Author

Joerg Arndt, Oct 27 2012

Keywords

• nonn

Comments

Parts are even, odd, even, odd, ... .

Examples

			The a(23) = 13 such partitions of 23 are:
[ 1]  2 3 18
[ 2]  2 5 16
[ 3]  2 7 14
[ 4]  2 9 12
[ 5]  2 21
[ 6]  4 5 14
[ 7]  4 7 12
[ 8]  4 9 10
[ 9]  4 19
[10]  6 7 10
[11]  6 17
[12]  8 15
[13]  10 13
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A179049 (parts are odd, even, odd, even, ...).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 08 2012; revised Feb 24 2020

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1-Mod[t, 2], If[i<1, 0, b[n, i-1, t] + If[i <= n && Mod[i, 2] != t, b[n-i, i-1, Mod[i, 2]], 0]]]; a[n_] := If[n==0, 1, Sum[ b[n-i, i-1, Mod[i, 2]], {i, 1, n}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

PARI

N=76; x='x+O('x^N);
gf179080 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n+1, 1-x^(2*k) ) );
gf179049 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n, 1-x^(2*k) ) );
gf = gf179080 - gf179049;
Vec( gf )

PARI

N=75; x='x+O('x^N); gf = sum(n=0, N, x^((n+1)*(n+4)/2) / prod(k=1, n+1, 1-x^(2*k) ) ); v2=Vec( gf )

Sage

# After Alois P. Heinz.
def A218355(n):
    @cached_function
    def h(n, k):
        if n == 0: return 1
        if k  > n: return 0
        return h(n, k+2) + h(n-k, k+1)
    return h(n, 2)
print([A218355(n) for n in range(76)]) # Peter Luschny, Feb 25 2020

Formula

G.f.: sum(n>=0, x^((n+1)*(n+4)/2) / prod(k=1..n+1, 1-x^(2*k) ) ).

a(n) = A179080(n) - A179049(n).