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a(n) = (n+1)!*Sum_{k=0..n} C(2k, k)*B(n-k), where B(n) is n-th Bernoulli number.
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%I #25 Sep 21 2015 19:27:48

%S 1,3,31,416,7316,158592,4079832,121418880,4102640064,155127605760,

%T 6488944560000,297483185986560,14831664692912640,798949604318423040,

%U 46240823333993702400,2861614126455843225600,188557593322666066329600

%N a(n) = (n+1)!*Sum_{k=0..n} C(2k, k)*B(n-k), where B(n) is n-th Bernoulli number.

%C The 1st negative term is a(64) = -1461516... (130 digits).

%C It appears that for n >= 64, a(n) < 0 if and only if n == 0 or 1 (mod 4). - _Robert Israel_, Sep 21 2015

%H Harry J. Smith and Robert Israel, <a href="/A061053/b061053.txt">Table of n, a(n) for n = 0..280</a> (n = 0..100 from Harry J. Smith)

%e a(3) = 4! *(binomial(0,0) B_3 + binomial(2,1) B_2 + binomial(4,2) B_1 + binomial(6,3) B_0) = 24 *(1 *0 + 2 *(1/6) + 6 *(-1/2) + 20 *1) = 416.

%p f:= n -> (n+1)!*add(binomial(2*k,k)*bernoulli(n-k),k=0..n):

%p map(f, [$0..100]); # _Robert Israel_, Sep 21 2015

%t Table[(n + 1)! Sum[Binomial[2 k, k] BernoulliB[n - k], {k, 0, n}], {n,

%t 0, 16}] (* _Michael De Vlieger_, Sep 21 2015 *)

%o (PARI) { default(realprecision, 500); for (n=0, 100, a=(n + 1)!*sum(k=0, n, binomial(2*k, k)*bernreal(n - k)); write("b061053.txt", n, " ", round(a)) ) } \\ _Harry J. Smith_, Jul 17 2009

%K easy,sign

%O 0,2

%A _Leroy Quet_, May 26 2001