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A368655 Binomial transform of Gould's sequence (A001316).

%I A368655 #31 Apr 26 2024 05:42:40
%S A368655 1,3,7,17,39,85,181,387,839,1829,3953,8391,17461,35759,72559,146921,
%T A368655 298631,611733,1265185,2641351,5555729,11735571,24798755,52219493,
%U A368655 109213269,226322799,464125219,941694917,1891879215,3769497853,7465462669,14735667195,29070011399
%N A368655 Binomial transform of Gould's sequence (A001316).
%C A368655 Consider the multivariate polynomial quotient ring K'_n = Z[x_1, x_2, x_3, ..., x_n]/I where I = <x_1^2 - P_1, x_2^2 - P_2, ..., x_n^2 - P_n> is an ideal in Z[x_1, x_2, x_3, ..., x_n]. Here, each polynomial P_i = -2x_i + x_{i+1} for 0 < i <= n, with x_{n+1} assumed to be 1. In this ring, every variable x_i for 0 < i <= n satisfies the recursive relation x_i^2 = -2x_i + x_{i+1}. The n-th term of this sequence is obtained by expanding the polynomial (2 + x_1)^n within the ring K'_n and evaluating at x_1 = x_2 = ... = x_n = 1. For a detailed explanation and proof, refer to Shunia's paper under links.
%H A368655 Joseph M. Shunia, <a href="/A368655/b368655.txt">Table of n, a(n) for n = 0..1000</a>
%H A368655 Vaclav Kotesovec, <a href="/A368655/a368655.jpg">Plot of a(n)/a(n-1) for n = 2..2000</a>
%H A368655 Joseph M. Shunia, <a href="https://arxiv.org/abs/2312.00302">A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence</a>, arXiv:2312.00302 [math.GM], 2023.
%F A368655 a(n) = Sum_{k=0..n} binomial(n,k)*A001316(k).
%F A368655 a(n) = Sum_{k=0..n} binomial(n,k)*2^(A000120(k)).
%t A368655 Table[Sum[Binomial[n, k] * 2^DigitCount[k, 2, 1], {k, 0, n}], {n, 0, 32}] (* _Vaclav Kotesovec_, Apr 02 2024 *)
%o A368655 (PARI) {a(n) = sum(k=0, n, binomial(n,k) * 2^hammingweight(k))};
%o A368655 (Sage)
%o A368655 def a(n):
%o A368655     R = PolynomialRing(ZZ, n, 'x')
%o A368655     x = R.gens()
%o A368655     I_list = [x[i]^2 - (-2*x[i] + x[i+1]) if i < n-1 else x[i]^2 for i in range(n)]
%o A368655     I = R.ideal(I_list)
%o A368655     K_n = R.quotient(I, 'x')
%o A368655     p_n = K_n((x[0]+2)^n)
%o A368655     subs_dict = {x[i]: 1 for i in range(n)}
%o A368655     a_n = p_n.lift().subs(subs_dict)
%o A368655     return a_n # _Joseph M. Shunia_, Mar 22 2024
%Y A368655 Cf. A000120, A001316 (Gould's sequence).
%K A368655 nonn,easy
%O A368655 0,2
%A A368655 _Joseph M. Shunia_, Jan 02 2024