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A130168
a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n).
4
1, 3, 15, 111, 1131, 15123, 256335, 5364471, 135751731, 4084163643, 144039790455, 5884504366431, 275643776229531, 14673941326078563, 880908054392169375, 59226468571935857991, 4432461082611507366531, 367227420727722013775883, 33514867695588319595233095
OFFSET
2,2
COMMENTS
As remarked by Gessel, A000366 has a combinatorial interpretation via a certain 2n X n array; this sequence is for a similar array of size (2n-1) X (n-1).
In effect, Dellac gives a combinatorial reason why the elements of A000366 are alternately -1 and +1 modulo 3. Dellac also shows that all the terms of this sequence are odd.
LINKS
Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file]
FORMULA
G.f.: 2*(1+x)/(3*x^3)*Q(0) - 2/(3*x) - 1/x^2 - 2/(3*x^3), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
a(n) = |(2-2^(n+2))*Bernoulli(n+1) - (n+1)*(1-2^(2n+2))*Bernoulli(2n+2) - (1-2^(2n+3))*Bernoulli(2n+3) + Sum_{k=0..n-1} (2*binomial(n,k+1)-binomial(n+1,k))*(1-2^(n+k+2))*Bernoulli(n+k+2)|/(3*2^(n-1)). - Chai Wah Wu, Apr 14 2023
MATHEMATICA
b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
a[n_] := (b[n] + b[n+1])/3;
a /@ Range[2, 20] (* Jean-François Alcover, Apr 08 2021 *)
PROG
(Python)
from math import comb
from sympy import bernoulli
def A130168(n): return (abs((2-(2<<n+1))*bernoulli(n+1)-(n+1)*(1-(1<<(m:=n+1<<1)))*bernoulli(m)-(1-(1<<m+1))*bernoulli(m+1)+sum((2*comb(n, k+1)-comb(n+1, k))*(1-(1<<(m:=n+k+2)))*bernoulli(m) for k in range(0, n)))>>n-1)//3 # Chai Wah Wu, Apr 14 2023
CROSSREFS
Sequence in context: A254789 A112936 A001063 * A267083 A089945 A135083
KEYWORD
nonn
AUTHOR
Don Knuth, Aug 02 2007
STATUS
approved