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A048998 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.

%I A048998 #28 Jun 20 2023 08:41:17
%S A048998 1,-1,2,1,-6,6,0,12,-36,24,-4,0,120,-240,120,0,-120,0,1200,-1800,720,
%T A048998 120,0,-2520,0,12600,-15120,5040,0,6720,0,-47040,0,141120,-141120,
%U A048998 40320,-12096,0,241920,0,-846720,0,1693440,-1451520,362880
%N A048998 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.
%C A048998 See A074909 for generators for the Bernoulli polynomials and connections to the beheaded Pascal triangle and reciprocals of the integers. - _Tom Copeland_, Nov 17 2014
%D A048998 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.
%H A048998 T. D. Noe, <a href="/A048998/b048998.txt">Rows n=0..50 of triangle, flattened</a>
%H A048998 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A048998 t*exp(x*t)/(exp(t)-1) = Sum_{n >= 0} B_n(x)*t^n/n!.
%F A048998 a(n,m) = [x^m]((n+1)!*B_n(x)), n>=0, m=0,...,n. - _Wolfdieter Lang_, Jun 21 2011
%e A048998 B_0(x)=1; B_1(x)=x-1/2; B_2(x)=x^2-x+1/6; B_3(x)=x^3-3*x^2/2+x/2; B_4(x)=x^4-2*x^3+x^2-1/30; ...
%e A048998 Triangle starts:
%e A048998    1;
%e A048998   -1,  2;
%e A048998    1, -6,   6;
%e A048998    0, 12, -36, 24;
%e A048998   ...
%p A048998 A048998 := proc(n,k) coeftayl(bernoulli(n,x),x=0,k) ; (n+1)!*% ; end proc:
%p A048998 seq(seq(A048998(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Jun 27 2011
%p A048998 # second program:
%p A048998 b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
%p A048998 (n + 1) * ((m + 1)*b(n - 1, m + 1, x) - (m + 1 - x)*b(n - 1, m, x)) fi end:
%p A048998 row := n -> seq(coeff(b(n, 0, x), x, k), k = 0..n):
%p A048998 seq(row(n), n = 0..8); # _Peter Luschny_, Jun 20 2023
%t A048998 Flatten[Table[CoefficientList[(n + 1)! BernoulliB[n, x], x], {n, 0, 10}]] (* _T. D. Noe_, Jun 21 2011 *)
%Y A048998 Cf. A048999, A074909.
%K A048998 sign,easy,nice,tabl
%O A048998 0,3
%A A048998 _N. J. A. Sloane_
%E A048998 Added 'Rising powers of x' in name - _Wolfdieter Lang_, Jun 21 2011