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