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

0,10

Comments

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

Links

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

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).

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A355001 A337713 A309425 * A103790 A249947 A193583

Adjacent sequences: A218352 A218353 A218354 * A218356 A218357 A218358

Keyword

nonn

Author

Joerg Arndt, Oct 27 2012

Status

approved