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

1, 3, 7, 17, 39, 85, 181, 387, 839, 1829, 3953, 8391, 17461, 35759, 72559, 146921, 298631, 611733, 1265185, 2641351, 5555729, 11735571, 24798755, 52219493, 109213269, 226322799, 464125219, 941694917, 1891879215, 3769497853, 7465462669, 14735667195, 29070011399

Offset

0,2

Comments

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.

Links

Joseph M. Shunia, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Plot of a(n)/a(n-1) for n = 2..2000

Joseph M. Shunia, A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence, arXiv:2312.00302 [math.GM], 2023.

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*A001316(k).

a(n) = Sum_{k=0..n} binomial(n,k)*2^(A000120(k)).

Mathematica

Table[Sum[Binomial[n, k] * 2^DigitCount[k, 2, 1], {k, 0, n}], {n, 0, 32}] (* Vaclav Kotesovec, Apr 02 2024 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * 2^hammingweight(k))};
(Sage)
def a(n):
    R = PolynomialRing(ZZ, n, 'x')
    x = R.gens()
    I_list = [x[i]^2 - (-2*x[i] + x[i+1]) if i < n-1 else x[i]^2 for i in range(n)]
    I = R.ideal(I_list)
    K_n = R.quotient(I, 'x')
    p_n = K_n((x[0]+2)^n)
    subs_dict = {x[i]: 1 for i in range(n)}
    a_n = p_n.lift().subs(subs_dict)
    return a_n # Joseph M. Shunia, Mar 22 2024

Crossrefs

Cf. A000120, A001316 (Gould's sequence).

Sequence in context: A176502 A319003 A141199 * A003478 A119587 A127984

Adjacent sequences: A368652 A368653 A368654 * A368656 A368657 A368658

Keyword

nonn,easy,changed

Author

Joseph M. Shunia, Jan 02 2024

Status

approved