A276425 - OEIS

%I #14 Oct 24 2016 05:58:55

%S 0,0,2,2,6,8,20,26,48,66,114,154,240,326,490,656,940,1252,1752,2306,

%T 3142,4104,5500,7114,9372,12030,15656,19932,25628,32402,41270,51816,

%U 65400,81608,102226,126800,157698,194550,240454,295110,362600,442902,541342,658230

%N Sum of the even singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

%H Alois P. Heinz, <a href="/A276425/b276425.txt">Table of n, a(n) for n = 0..2000</a>

%F G.f.:  g(x) = 2x^2*(1+x^2+x^4)/((1-x^4)^2 product(1-x^j, j>=1)).

%F a(n) = Sum(k*A276424(n,k), k>=0).

%e a(4) = 6 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively; their sum is 6.

%e a(5) = 8 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0 respectively; their sum is 8.

%p g := 2*x^2*(1+x^2+x^4)/((1-x^4)^2*(product(1-x^i, i = 1 .. 120))): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 50);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p       `if`(i<1, 0, add((p-> p+`if`(i::even and j=1,

%p       [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 14 2016

%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, 0, Sum[Function[p, p + If[EvenQ[i] && j == 1, {0, i*p[[1]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Oct 24 2016, after _Alois P. Heinz_ *)

%Y Cf. A103628, A276422, A276423, A276424.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Sep 14 2016