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
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
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
KEYWORD
AUTHOR
R. J. Mathar, Oct 14 2011
STATUS
approved