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A129378 Row sums of coefficients of Bernoulli twin number polynomials. 5
1, 1, 4, 20, 116, 744, 5160, 39360, 350784, 3749760, 42940800, 442713600, 4650877440, 109244298240, 2833294464000, -3487131648000, -2166903606067200, 51809012320665600, 6808619561103360000, -131306587205713920000, -26982365129174827008000, 595860034297401409536000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..21.

FORMULA

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

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-Fran├žois Alcover, Aug 08 2012, after given formula *)

CROSSREFS

Cf. A129724, A129826.

Sequence in context: A165311 A100328 A082298 * A078944 A158900 A190194

Adjacent sequences:  A129375 A129376 A129377 * A129379 A129380 A129381

KEYWORD

sign

AUTHOR

Paul Curtz, Jun 08 2007

EXTENSIONS

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

STATUS

approved

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Last modified September 18 23:16 EDT 2021. Contains 347548 sequences. (Running on oeis4.)