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%I #24 Feb 25 2020 05:47:03
%S 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,
%T 23,25,29,30,37,35,46,41,58,49,70,57,85,68,103,81,123,97,145,115,172,
%U 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
%N Number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is even.
%C Parts are even, odd, even, odd, ... .
%H Alois P. Heinz, <a href="/A218355/b218355.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: sum(n>=0, x^((n+1)*(n+4)/2) / prod(k=1..n+1, 1-x^(2*k) ) ).
%F a(n) = A179080(n) - A179049(n).
%e The a(23) = 13 such partitions of 23 are:
%e [ 1] 2 3 18
%e [ 2] 2 5 16
%e [ 3] 2 7 14
%e [ 4] 2 9 12
%e [ 5] 2 21
%e [ 6] 4 5 14
%e [ 7] 4 7 12
%e [ 8] 4 9 10
%e [ 9] 4 19
%e [10] 6 7 10
%e [11] 6 17
%e [12] 8 15
%e [13] 10 13
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
%p end:
%p a:= n-> b(n, 2):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Nov 08 2012; revised Feb 24 2020
%t 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_ *)
%o (PARI)
%o N=76; x='x+O('x^N);
%o gf179080 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n+1, 1-x^(2*k) ) );
%o gf179049 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n, 1-x^(2*k) ) );
%o gf = gf179080 - gf179049;
%o Vec( gf )
%o (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 )
%o (Sage) # After _Alois P. Heinz_.
%o def A218355(n):
%o @cached_function
%o def h(n, k):
%o if n == 0: return 1
%o if k > n: return 0
%o return h(n, k+2) + h(n-k, k+1)
%o return h(n, 2)
%o print([A218355(n) for n in range(76)]) # _Peter Luschny_, Feb 25 2020
%Y Cf. A179049 (parts are odd, even, odd, even, ...).
%K nonn
%O 0,10
%A _Joerg Arndt_, Oct 27 2012
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