%I #21 Feb 03 2024 10:13:19
%S 1,1,4,20,116,744,5160,39360,350784,3749760,42940800,442713600,
%T 4650877440,109244298240,2833294464000,-3487131648000,
%U -2166903606067200,51809012320665600,6808619561103360000,-131306587205713920000,-26982365129174827008000,595860034297401409536000
%N Row sums of coefficients of Bernoulli twin number polynomials.
%C The origin of the sequence are polynomials on pages 61 and 69 of the CCSA paper. The first few of the polynomials have been noted in the 1992 Gazette paper.
%C We construct Bernoulli twin numbers polynomials C(n,x) = Sum_{j=1..n} binomial(n-1,j-1)*B(j,x) where B(n,x) are the Bernoulli polynomials of A048998 and A048999 and where binomial(.,.) is the Pascal triangle A007318: C(0,x)=B(0,x); C(1,x)=B(1,x); C(2,x)=B(2,x)+B(1,x); C(3,x)=B(3,x)+2B(2,x)+B(1,x).
%C The triangle of coefficients [x^m] C(n,x) for rows n=0,1,2,.. and decreasing power m=n,...,0 along each row starts
%C 1;
%C 1, -1/2;
%C 1, 0, -1/3;
%C 1, 1/2, -1/2, -1/6;
%C The rightmost fraction in row n, that is, the absolute term C(n,0), is the Bernoulli twin number C(n) of A129826(n), i.e., C(n) = A129826(n)/(n+1)!.
%C If rows are multiplied by (n+1)!, the triangle becomes
%C 1;
%C 2, -1;
%C 6, 0, -2;
%C 24, 12, -12, -4;
%C 120, 120, -60, -60, -4;
%C The sequence a(n) gives the row sums of this triangle. The sums of antidiagonals are 1, 2, 5, 24, 130, 828, 6056.... The first column of the inverse of the triangle is 1, 2, 3, 3, 0, (0 continued).
%D P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
%D P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
%H G. C. Greubel, <a href="/A129378/b129378.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.
%t c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1] = -1/2; c[2] = -1/3; a[n_] := (n+1)!*(1+c[n]); a[0]=1; Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 08 2012, after given formula *)
%o (Magma)
%o f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
%o A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
%o [A129378(n): n in [0..30]]; // _G. C. Greubel_, Feb 01 2024
%o (SageMath)
%o def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
%o def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
%o [A129378(n) for n in range(31)] # _G. C. Greubel_, Feb 01 2024
%Y Cf. A000142, A007318, A048998, A048999, A129724, A129826.
%K sign
%O 0,3
%A _Paul Curtz_, Jun 08 2007
%E Edited and extended by _R. J. Mathar_, Aug 06 2008