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A218355 Number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is even. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A250207 A216319 A309425 * A103790 A249947 A193583

Adjacent sequences:  A218352 A218353 A218354 * A218356 A218357 A218358

KEYWORD

nonn

AUTHOR

Joerg Arndt, Oct 27 2012

STATUS

approved

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Last modified August 3 08:53 EDT 2020. Contains 336197 sequences. (Running on oeis4.)