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A369628 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

%I A369628 #8 Jan 29 2024 19:27:33
%S A369628 0,1,2,3,6,15,36,85,213,549,1423,3723,9882,26508,71579,194533,532120,
%T A369628 1463561,4044075,11221727,31260192,87386579,245058185,689209348,
%U A369628 1943530845,5494106583,15566303698,44196212866,125727934145,358317169828,1022916667066,2924843243594
%N A369628 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = 1, where k_i are from {-1,0,1}, i=1..n.
%F A369628 a(n) = [x^1] Product_{k=1..n} (x^k + 1 + 1/x^k).
%F A369628 a(n) = [x^(n*(n+1)/2+1)] Product_{k=1..n} (1 + x^k + x^(2*k)).
%t A369628 Table[Coefficient[Product[(x^k + 1 + 1/x^k), {k, 1, n}], x, 1], {n, 0, 31}]
%o A369628 (Python)
%o A369628 from itertools import count, islice
%o A369628 from collections import Counter
%o A369628 def A369628_gen(): # generator of terms
%o A369628     ccount = Counter({0:1})
%o A369628     yield 0
%o A369628     for i in count(1):
%o A369628         bcount = Counter(ccount)
%o A369628         for a in ccount:
%o A369628             bcount[a+i] += ccount[a]
%o A369628             bcount[a-i] += ccount[a]
%o A369628         ccount = bcount
%o A369628         yield(ccount[1])
%o A369628 A369628_list = list(islice(A369628_gen(),20)) # _Chai Wah Wu_, Jan 29 2024
%Y A369628 Cf. A007576, A063866, A316706.
%K A369628 nonn
%O A369628 0,3
%A A369628 _Ilya Gutkovskiy_, Jan 28 2024