A239977 - OEIS

%I #44 Sep 08 2022 08:46:07

%S 0,1,3,6,9,10,9,14,33,18,-135,22,2097,26,-38199,30,929601,34,

%T -28820583,38,1109652945,42,-51943281687,46,2905151042529,50,

%U -191329672483911,54,14655626154768753,58,-1291885088448017655,62,129848163681107302017,66

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

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

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

%C 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).

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

%C    1,   2,   3,   3,   1,  -1, ...

%C    1,   1,   0,  -2,  -2,   6, ...

%C    0,  -1,  -2,   0,   8,   8, ...

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

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

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

%C 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

%H B. Sury, <a href="http://www.isibang.ac.in/~sury/bulllms.pdf">The value of Bernoulli Polynomials at rational numbers</a>, Bull. London Math. Soc. 25 (1993), 327-29.

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

%F a(2n) is divisible by 3.

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

%F a(n) = 2*((1 - 2^n)*B(n, 1) + n), B(n, x) the Bernoulli polynomial. - _Peter Luschny_, Apr 16 2014

%F From _Peter Bala_, Mar 02 2015: (Start)

%F 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.

%F 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).

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

%F a(n) = Sum_{k=0..n-1} 2^k*binomial(n, k)*Bernoulli(k, 1). - _Peter Luschny_, Aug 17 2021

%p A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):

%p seq(A239977(n), n=0..33); # _Peter Luschny_, Mar 08 2015

%t 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 *)

%o (Magma) [0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // _Vincenzo Librandi_, Mar 03 2015

%Y Cf. A083007.

%K sign,easy

%O 0,3

%A _Paul Curtz_, Mar 30 2014