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A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.

%I A259456 #44 Aug 05 2025 14:39:48
%S A259456 1,2,3,6,20,15,24,130,210,105,120,924,2380,2520,945,720,7308,26432,
%T A259456 44100,34650,10395,5040,64224,303660,705320,866250,540540,135135,
%U A259456 40320,623376,3678840,11098780,18858840,18288270,9459450,2027025,362880,6636960,47324376,177331440,389449060,520059540,416215800
%N A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.
%D A259456 L. Comtet, Advanced Combinatorics (1974), Chapter VI, page 256.
%D A259456 DJ Jeffrey, GA Kalugin, N Murdoch, Lagrange inversion and Lambert W, Preprint 2015; http://www.apmaths.uwo.ca/~djeffrey/Offprints/JeffreySYNASC2015paper17.pdf
%D A259456 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.
%H A259456 Lothar Berg, <a href="http://ftp.math.uni-rostock.de/pub/romako/pdf_romako/Rahmen56.pdf#page=55">On polynomials related with generalized Bernoulli numbers</a>, Rostock Math. Kolloq. (2002).
%H A259456 Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. 13 (2010) #10.4.4 page 4.
%H A259456 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H A259456 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 6.
%H A259456 Donald E. Knuth, <a href="http://www.mathematica-journal.com/issue/v2i4/article/knuth/index.html">Convolution polynomials</a>, The Mathematica J., 2 (1992), 67-78.
%H A259456 Donald E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA], 1992.
%H A259456 Richard B. Paris, <a href="http://dx.doi.org/10.1016/j.cam.2006.11.017">An asymptotic approximation for incomplete Gaussian sums. II.</a>, J. Comp. Appl. Math 212 (2008) 16-30, Table 1.
%H A259456 Grzegorz Rzadkowski, <a href="http://arxiv.org/abs/1007.1955">On some expansions for the Euler Gamma function and the Riemann Zeta function</a>, arxiv:1007.1955 [math.CA], Table 1. J. Comp. Appl. Math. 236 (15) (2012), 3710-3719.
%H A259456 Lajos Takács, <a href="http://dx.doi.org/10.1137/0403050">On the number of distinct forests</a>, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives a version of the triangle.
%F A259456 T(n,k) = (n-k-1)*( T(n-1,k-1)+T(n-1,k) ), n>=1, 1<=k<=n. [Berg, Eq. 6]
%F A259456 The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - _Tom Copeland_, Sep 20 2016
%e A259456 Triangle begins:
%e A259456 1,
%e A259456 2,3,
%e A259456 6,20,15,
%e A259456 24,130,210,105,
%e A259456 120,924,2380,2520,945,
%e A259456 ...
%e A259456 For k=4 and j=2 in Knuth's equation, |S1(4,4-2)| = |S1(4,2)| = |A008275(4,2)| = 11 = p_{2,1}*C(4,3) +p_{2,2}*C(4,4) = 2*4+3*1. - _R. J. Mathar_, Jul 16 2015
%p A259456 A259456 := proc(n,k)
%p A259456     option remember;
%p A259456     if k < 1 or k > n  then
%p A259456         0 ;
%p A259456     elif n = 1 then
%p A259456         1;
%p A259456     else
%p A259456         procname(n-1,k-1)+procname(n-1,k);
%p A259456         %*(n+k-1) ;
%p A259456     end if;
%p A259456 end proc:
%p A259456 seq(seq(A259456(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Jul 18 2015
%t A259456 T[n_, k_] := T[n, k] = If[k < 1 || k > n, 0, If[n == 1, 1, (T[n-1, k-1] + T[n-1, k])(n+k-1)]];
%t A259456 Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2019, from Maple *)
%Y A259456 Cf. This is a row reversed and unsigned version of A111999.
%Y A259456 Cf. A008275, A000276 (2nd column), A000483 (3rd column), A000142 (1st column).
%Y A259456 Cf. A133932.
%K A259456 nonn,tabl,easy
%O A259456 0,2
%A A259456 _N. J. A. Sloane_, Jun 30 2015