A197419 Triangle with the numerator of the coefficient [x^k] of the second order Bernoulli polynomial B_n^(2)(x) in row n, column 0<=k<=n.

1, -1, 1, 5, -2, 1, -1, 5, -3, 1, 1, -2, 5, -4, 1, 1, 1, -5, 25, -5, 1, -5, 1, 3, -10, 25, -6, 1, -1, -5, 7, 7, -35, 35, -7, 1, 7, -4, -10, 28, 7, -28, 70, -8, 1, 3, 21, -6, -10, 21, 63, -42, 30, -9, 1, -15, 3, 21, -20, -25, 42, 21, -60, 75, -10, 1, -5, -15, 33, 77, -55, -55, 77, 33, -165, 275, -11, 1, 7601, -10, -45, 66, 231, -132, -110, 132, 99, -110, 55, -12, 1

Offset

0,4

Comments

The a-th order Bernoulli polynomials are defined via the exponential generating function (t/(exp t -1))^a*exp(x*t) = sum_{n>=0} B_n^(a)(x) * t^n/n!. The current triangular array shows the coefficient [x^k] of B_n^(2)(x), i.e. the expansion coefficients in rising powers of the polynomial of x with a=2.

P(n,x) = 2*sum(m=0..n-1, binomial(n,m)*sum(k=1..n-m, stirling2(n-m,k) * stirling1(2+k,2)/((k+1)*(k+2))))*x^m+x^n. - Vladimir Kruchinin, Oct 23 2011]

Links

Table of n, a(n) for n=0..90.

R. Dere, Y. Simsek, Bernoulli type polynomials on Umbral Algebra, arXiv:1110.1484 [math.CA]

V. Kruchinin, D. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, arXiv: 1211.0099

Formula

T(n,m) = sum(2*C(n,m)*sum(k=1..n-m, stirling2(n-m,k)*stirling1(2+k,2)/ ((k+1)*(2+k)))), m<n, T(n,n)=1. - Vladimir Kruchinin, Oct 23 2011

Example

The table of the coefficients is
1;
-1,1;
5/6,-2,1;     5/6-2x+x^2
-1/2,5/2,-3,1;   -1/2+5x/2-3x^2+x^3
1/10,-2,5,-4,1;
1/6,1/2,-5,25/3,-5,1;
-5/42,1,3/2,-10,25/2,-6,1;
-1/6,-5/6,7/2,7/2,-35/2,35/2,-7,1;
7/30,-4/3,-10/3,28/3,7,-28,70/3,-8,1;
3/10,21/10,-6,-10,21,63/5,-42,30,-9,1;
-15/22,3,21/2,-20,-25,42,21,-60,75/2,-10,1;
-5/6,-15/2,33/2,77/2,-55,-55,77,33,-165/2,275/6,-11,1;
7601/2730,-10,-45,66,231/2,-132,-110,132,99/2,-110,55,-12,1;

Maple

A197419 := proc(n, k)
        local a, Bt, Bnx, o , t, x;
        a := 2 ;
        Bt := (t/(exp(t)-1))^a*exp(x*t) ;
        Bnx := n!*coeftayl(Bt, t=0, n) ;
        coeftayl(Bnx, x=0, k) ;
        numer(%) ;
end proc:
seq(seq(A197419(n, k), k=0..n), n=0..4) ; # print row by row

Mathematica

t[n_, m_] := If [n == m, 1, 2*Binomial[n, m]*Sum[StirlingS2[n-m, k]*StirlingS1[2+k, 2]/((k+1)*(2+k)), {k, 1, n-m}]]; Table[t[n, m] // Numerator, {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2013, after Vladimir Kruchinin *)

Prog

(Maxima) T(n, m):=num(if n=m then 1 else 2*binomial(n, m)* sum(stirling2(n-m, k) *stirling1(2+k, 2)/ ((k+1)*(2+k)), k, 1, n-m)); [From Vladimir Kruchinin, Oct 23 2011]

Crossrefs

Cf. A197420 (denominator), A100616, A100615 (column k=0).

Sequence in context: A010128 A348993 A180133 * A029764 A136301 A132690

Adjacent sequences: A197416 A197417 A197418 * A197420 A197421 A197422

Keyword

sign,tabl,frac

Author

R. J. Mathar, Oct 14 2011

Status

approved