This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129378 #21 Feb 03 2024 10:13:19
%S A129378 1,1,4,20,116,744,5160,39360,350784,3749760,42940800,442713600,
%T A129378 4650877440,109244298240,2833294464000,-3487131648000,
%U A129378 -2166903606067200,51809012320665600,6808619561103360000,-131306587205713920000,-26982365129174827008000,595860034297401409536000
%N A129378 Row sums of coefficients of Bernoulli twin number polynomials.
%C A129378 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 A129378 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 A129378 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 A129378 1;
%C A129378 1, -1/2;
%C A129378 1, 0, -1/3;
%C A129378 1, 1/2, -1/2, -1/6;
%C A129378 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 A129378 If rows are multiplied by (n+1)!, the triangle becomes
%C A129378 1;
%C A129378 2, -1;
%C A129378 6, 0, -2;
%C A129378 24, 12, -12, -4;
%C A129378 120, 120, -60, -60, -4;
%C A129378 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 A129378 P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
%D A129378 P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
%H A129378 G. C. Greubel, <a href="/A129378/b129378.txt">Table of n, a(n) for n = 0..300</a>
%F A129378 a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.
%t A129378 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 A129378 (Magma)
%o A129378 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 A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
%o A129378 [A129378(n): n in [0..30]]; // _G. C. Greubel_, Feb 01 2024
%o A129378 (SageMath)
%o A129378 def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
%o A129378 def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
%o A129378 [A129378(n) for n in range(31)] # _G. C. Greubel_, Feb 01 2024
%Y A129378 Cf. A000142, A007318, A048998, A048999, A129724, A129826.
%K A129378 sign
%O A129378 0,3
%A A129378 _Paul Curtz_, Jun 08 2007
%E A129378 Edited and extended by _R. J. Mathar_, Aug 06 2008