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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.

%I A197419 #36 Feb 16 2025 05:41:01
%S A197419 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,
%T A197419 1,-1,-5,7,7,-35,35,-7,1,7,-4,-10,28,7,-28,70,-8,1,3,21,-6,-10,21,63,
%U A197419 -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
%N 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.
%C A197419 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.
%C A197419 P(n,x) = x^n + 2*Sum_{m=0..n-1} binomial(n,m)*x^m*Sum_{k=1..n-m} stirling2(n-m,k)*stirling1(2+k,2)/((k+1)*(k+2)). - _Vladimir Kruchinin_, Oct 23 2011
%H A197419 R. Dere and Y. Simsek, <a href="http://arxiv.org/abs/1110.1484">Bernoulli type polynomials on Umbral Algebra</a>, arXiv:1110.1484 [math.CA], 2011.
%H A197419 Vladimir Kruchinin and D. Kruchinin, <a href="http://arxiv.org/abs/1211.0099">Application of a composition of generating functions for obtaining explicit formulas of polynomials</a>, arXiv: 1211.0099 [math.NT], 2012.
%F A197419 T(n,m) = 2*binomial(n,m)*Sum_{k=1..n-m} Stirling2(n-m,k)*Stirling1(2+k,2)/((k+1)*(k+2)), m<n, T(n,n)=1. - _Vladimir Kruchinin_, Oct 23 2011
%e A197419 The table of the coefficients is
%e A197419   1;
%e A197419   -1,1;
%e A197419   5/6,-2,1;     5/6-2x+x^2
%e A197419   -1/2,5/2,-3,1;   -1/2+5x/2-3x^2+x^3
%e A197419   1/10,-2,5,-4,1;
%e A197419   1/6,1/2,-5,25/3,-5,1;
%e A197419   -5/42,1,3/2,-10,25/2,-6,1;
%e A197419   -1/6,-5/6,7/2,7/2,-35/2,35/2,-7,1;
%e A197419   7/30,-4/3,-10/3,28/3,7,-28,70/3,-8,1;
%e A197419   3/10,21/10,-6,-10,21,63/5,-42,30,-9,1;
%e A197419   -15/22,3,21/2,-20,-25,42,21,-60,75/2,-10,1;
%e A197419   -5/6,-15/2,33/2,77/2,-55,-55,77,33,-165/2,275/6,-11,1;
%e A197419   7601/2730,-10,-45,66,231/2,-132,-110,132,99/2,-110,55,-12,1;
%p A197419 A197419 := proc(n,k)
%p A197419         local a,Bt,Bnx,o ,t,x;
%p A197419         a := 2 ;
%p A197419         Bt := (t/(exp(t)-1))^a*exp(x*t) ;
%p A197419         Bnx := n!*coeftayl(Bt,t=0,n) ;
%p A197419         coeftayl(Bnx,x=0,k) ;
%p A197419         numer(%) ;
%p A197419 end proc:
%p A197419 seq(seq(A197419(n,k),k=0..n),n=0..4) ; # print row by row
%t A197419 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_ *)
%o A197419 (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)); /* _Vladimir Kruchinin_, Oct 23 2011 */
%Y A197419 Cf. A197420 (denominator), A100616, A100615 (column k=0).
%K A197419 sign,tabl,frac
%O A197419 0,4
%A A197419 _R. J. Mathar_, Oct 14 2011