A069994 a(n) = Sum_{i=0..2n} B(i)*C(2n+1,i)*6^i where B(i) are the Bernoulli numbers, C(2n,i) the binomial coefficients.

-2, 10, -170, 6370, -415826, 41649850, -5922729722, 1134081384850, -281284596509858, 87722769712529770, -33597252908389628234, 15502327024398065811010, -8481855507605264686660850, 5429636257086663655134162970

Offset

1,1

Comments

Related to those formulas derived from Bernoulli polynomials: Sum_{k>0} sin(k*x)/k^(2n+1) = (-1)^(n+1)/2*x^(2n+1)/(2n+1)!*Sum_{i=0..2n} (2Pi/x)^i*B(i)*C(2n+1,i).

Links

Table of n, a(n) for n=1..14.

Wikipedia, Bernoulli Polynomials

Formula

From Peter Bala, Mar 02 2015: (Start)

a(n) = 6^(2*n - 1)*B(2*n - 1,1/6), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A002111, A009843, A069852.

Conjecturally, a(n) = 2 * the unsigned numerator of B(2*n - 1,1/6). If true then this sequence is a bisection of 2*A158073.

G.f.: -3*t*sinh(2*t)/sinh(3*t) = -2*t + 10*t^3/3! - 170*t^5/5! + ....

G.f.: Sum_{n >= 0} { 2/(n+1) * Sum_{k = 0..n} (-1)^(k+1)*binomial(n,k)/( (1 - (6*k + 1)*x)*(1 - (6*k + 5)*x) ) } = -2 + 10*x^2 - 170*x^4 + 6370*x^6 - ....

(End)

Maple

seq(6^(2*n-1)*bernoulli(2*n-1, 1/6), n=1..14); # (after Peter Bala) Peter Luschny, Mar 08 2015

Prog

(PARI) for(n=1, 25, print1(sum(i=0, 2*n, binomial(2*n+1, i)*bernfrac(i)*6^i), ", "))

Crossrefs

Cf. A002111, A009843, A069852, A158073.

Sequence in context: A126451 A260122 A132341 * A063573 A368573 A356887

Adjacent sequences: A069991 A069992 A069993 * A069995 A069996 A069997

Keyword

easy,sign

Author

Benoit Cloitre, May 01 2002

Status

approved