A395257 - OEIS

%I #28 May 16 2026 23:41:43

%S 0,0,0,1,0,1,2,3,4,6,7,10,13,18,22,29,34,43,51,63,74,90,104,124,143,

%T 167,191,222,251,288,325,369,413,467,519,582,645,718,792,878,963,1062,

%U 1162,1275,1389,1519,1649,1796,1945,2110,2278,2465,2653,2862,3074,3306,3542,3801,4063

%N Number of strict integer partitions of 2*n with reverse-alternating sum 6.

%C Also the number of partitions of n+18 into 6 distinct parts containing the part 6.

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

%F G.f.: Sum_{j=1..6} q^(j*(j-1)+3) * q_binomial(5,j-1) / Product_{k=1..j-1} (1-q^k).

%F G.f.: Sum_{j=1..6} (-1)^(j-1) * q^(j*(j-1)/2+3) / Product_{k=1..6-j} (1-q^k).

%F a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n-15) for n > 33.

%o (PARI) my(N=60, q='q+O('q^N)); concat([0, 0, 0], Vec(sum(j=1, 6, (-1)^(j-1)*q^(j*(j-1)/2+3)/prod(k=1, 6-j, 1-q^k))))

%Y Column k=3 of A344649.

%Y Cf. A395222.

%K nonn

%O 0,7

%A _Seiichi Manyama_, May 16 2026