A027196 - OEIS

%I #17 May 15 2023 11:13:54

%S 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,4,5,6,8,9,12,14,17,20,25,29,35,

%T 41,50,58,70,81,97,113,134,156,185,214,252,292,343,396,463,534,623,

%U 718,833,958,1110,1274,1471,1686,1943,2223,2555,2919,3347,3818,4368

%N Number of partitions of n into an even number of parts, the least being 4; also, a(n+4) = number of partitions of n into an odd number of parts, each >=4.

%F a(n) +  A027190(n) = A026797(n). - _R. J. Mathar_, Oct 18 2019

%F G.f.: x^8 * Sum_{k>=0} x^(8*k)/Product_{j=1..2*k+1} (1-x^j). - _Seiichi Manyama_, May 15 2023

%p b:= proc(n, i, t) option remember; `if`(n=0, t,

%p      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))

%p     end:

%p a:= n-> `if`(n<4, 0, b(n-4, 4, 0)):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 18 2019

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];

%t a[n_] := If[n < 4, 0, b[n - 4, 4, 0]];

%t Array[a, 100] (* _Jean-François Alcover_, May 17 2020, after _Alois P. Heinz_ *)

%Y Cf. A026797, A027190.

%K nonn

%O 1,16

%A _Clark Kimberling_