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A369496 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

%I A369496 #12 Apr 26 2025 20:52:59
%S A369496 1,1,1,1,2,4,8,20,47,104,246,607,1496,3751,9579,24720,64327,168932,
%T A369496 446830,1188030,3177198,8541152,23063100,62550085,170337684,465564180,
%U A369496 1276779917,3512617527,9692054125,26815357935,74381739478,206820705565,576371104028
%N A369496 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).
%H A369496 Alois P. Heinz, <a href="/A369496/b369496.txt">Table of n, a(n) for n = 0..200</a>
%p A369496 b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
%p A369496      `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+b(n, i-1)+
%p A369496         b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
%p A369496     end:
%p A369496 a:= n-> b(n*(n+1)/2, n):
%p A369496 seq(a(n), n=0..32);  # _Alois P. Heinz_, Jan 24 2024
%t A369496 Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, n (n + 1)/2], {n, 0, 32}]
%Y A369496 Cf. A000217, A316706, A369344, A369434.
%K A369496 nonn
%O A369496 0,5
%A A369496 _Ilya Gutkovskiy_, Jan 24 2024