A129378 Row sums of coefficients of Bernoulli twin number polynomials.

1, 1, 4, 20, 116, 744, 5160, 39360, 350784, 3749760, 42940800, 442713600, 4650877440, 109244298240, 2833294464000, -3487131648000, -2166903606067200, 51809012320665600, 6808619561103360000, -131306587205713920000, -26982365129174827008000, 595860034297401409536000

Offset: 0

Author

Paul Curtz, Jun 08 2007

Keywords

• sign

Comments

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.

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).

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

1;

1, -1/2;

1, 0, -1/3;

1, 1/2, -1/2, -1/6;

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)!.

If rows are multiplied by (n+1)!, the triangle becomes

1;

2, -1;

6, 0, -2;

24, 12, -12, -4;

120, 120, -60, -60, -4;

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).

References

• P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
• P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000142, A007318, A048998, A048999, A129724, A129826.

Programs

Magma

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
[A129378(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

Mathematica

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 *)

SageMath

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
[A129378(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

Formula

a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.

Extensions

Edited and extended by R. J. Mathar, Aug 06 2008