A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

0, 1, 3, 6, 9, 10, 9, 14, 33, 18, -135, 22, 2097, 26, -38199, 30, 929601, 34, -28820583, 38, 1109652945, 42, -51943281687, 46, 2905151042529, 50, -191329672483911, 54, 14655626154768753, 58, -1291885088448017655, 62, 129848163681107302017, 66

Offset: 0

Author

Paul Curtz, Mar 30 2014

Keywords

• sign
• easy

Comments

Let T(n, k) denote the difference table of a(n).

(-1)^(k+1)*T(3, 3+k) = T(k+3, 3) for k >= 0.

Without the first two rows and the first two columns we have the core of the Genocchi numbers, like A240581(n)/A239315(n) for the Bernoulli numbers. See A226158(n).

0, 1, 3, 6, 9, 10, ...

1, 2, 3, 3, 1, -1, ...

1, 1, 0, -2, -2, 6, ...

0, -1, -2, 0, 8, 8, ...

-1, -1, 2, 8, 0, -56, ...

0, 3, 6, -8, -56, 0, ...

The definition reflects the identity 2*((1-2^n)*B(n,1) + n) =

2*Sum_{k=0..n} C(n,k)*(2^k-1)*B(k,1) where B(n,x) denotes the Bernoulli polynomials. - Peter Luschny, Apr 16 2014

Links

• B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.

Crossrefs

Cf. A083007.

Programs

Magma

[0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // Vincenzo Librandi, Mar 03 2015

Maple

A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):
seq(A239977(n), n=0..33); # Peter Luschny, Mar 08 2015

Mathematica

a[n_] := (EulerE[n-1, 0]+2)*n; a[0] = 0; a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2014 *)

Formula

a(n) = 2*n + A226158(n).

a(2n) is divisible by 3.

a(2n+1) = A133653(n).

a(n) = 2*((1 - 2^n)*B(n, 1) + n), B(n, x) the Bernoulli polynomial. - Peter Luschny, Apr 16 2014

From Peter Bala, Mar 02 2015: (Start)

a(n) = (-2)^n * ( B(n,-1/2) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, for any nonzero integer k, k^n*( B(n,1/k) - B(n,0) ) is an integer for n >= 0. Cf. A083007.

a(0) = 0 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} (-2)^(n-k)*binomial(n+1,k)*a(k).

E.g.f.: 2*x*exp(2*x)/(1 + exp(x)) = x + 3*x^2/2! + 6*x^3/3! + .... (End)

a(n) = Sum_{k=0..n-1} 2^k*binomial(n, k)*Bernoulli(k, 1). - Peter Luschny, Aug 17 2021