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A137375 Triangle read by rows, T(n,k) = (-1)^k*{{n,k}} where {{n,k}} are the second-order Stirling set numbers, n>=0, 0<=k<=n/2.

%I A137375 #40 Feb 16 2025 08:33:07
%S A137375 1,0,0,-1,0,-1,0,-1,3,0,-1,10,0,-1,25,-15,0,-1,56,-105,0,-1,119,-490,
%T A137375 105,0,-1,246,-1918,1260,0,-1,501,-6825,9450,-945,0,-1,1012,-22935,
%U A137375 56980,-17325,0,-1,2035,-74316,302995,-190575,10395,0,-1,4082,-235092
%N A137375 Triangle read by rows, T(n,k) = (-1)^k*{{n,k}} where {{n,k}} are the second-order Stirling set numbers, n>=0, 0<=k<=n/2.
%C A137375 Also called "associated Stirling numbers of the second kind" by Riordan in the signless form with different offset A008299.
%C A137375 Old name was: Triangular sequence from coefficients of Mahler polynomials from expansion of: p(x) = exp(x*(1 + t - exp(t))) with weight n!:M(x,n).
%C A137375 M(n,x) = sum(k=0..n, (x)^k*sum(j=0..k, binomial(n,k-j)*stirling2(n-k+j,j)*(-1)^j)). - _Vladimir Kruchinin_, Jan 13 2012
%D A137375 J. Riordan, Introduction to Combinatory Analysis, Wiley, New York, 1958.
%H A137375 L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1965-0172814-6">The coefficients in an asymptotic expansion</a>, Proc. Amer. Math. Soc. 16 (1965) 248-252.
%H A137375 A. E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%H A137375 V. Kruchinin, 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.
%H A137375 D. V. Kruchinin and V. V. Kruchinin, <a href="http://dx.doi.org/10.12785/amis/070550">Explicit Formulas for Some Generalized Polynomials</a>, Appl. Math. Inf. Sci. 7, No. 5, 2083-2088 (2013).
%H A137375 Andrew Elvey Price, Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H A137375 L. M. Smiley, <a href="http://arxiv.org/abs/math/0006106">Completion of a rational function sequence of Carlitz</a>, arXiv:math/0006106 [math.CO], 2000.
%H A137375 E. W. Weisstein, <a href="https://mathworld.wolfram.com/MahlerPolynomial.html">Mahler Polynomial</a>.
%F A137375 T(n,k) = Sum_{j=0..k} C(n,k-j)*stirling2(n-k+j,j)*(-1)^(j). - _Vladimir Kruchinin_, Jan 13 2012
%F A137375 T(n,k) = (-1)^k*Sum_{j=0..n-k} C(j,n-2*k)*E2(n-k,n-k-j-1) for n>0, T(0,0) = 1, where E2(n,k) are the second-order Eulerian numbers A201637. - _Peter Luschny_, Nov 27 2012
%F A137375 Let p(x,t) = exp(x*(1+t-exp(t))) then T(n,k) = [x^k](n!*[t^n] series(p(x,t))) where [s^m] denotes the coefficient of s^m. - _Peter Luschny_, Dec 01 2012
%e A137375 [ 0]  1;
%e A137375 [ 1]  0;
%e A137375 [ 2]  0, -1;
%e A137375 [ 3]  0, -1;
%e A137375 [ 4]  0, -1,   3;
%e A137375 [ 5]  0, -1,  10;
%e A137375 [ 6]  0, -1,  25,   -15;
%e A137375 [ 7]  0, -1,  56,  -105;
%e A137375 [ 8]  0, -1, 119,  -490,  105;
%e A137375 [ 9]  0, -1, 246, -1918, 1260;
%e A137375 [10]  0, -1, 501, -6825, 9450, -945;
%p A137375 A137375 := proc(n, k) if n = 0 then 1 else
%p A137375 add(binomial(j,n-2*k)* combinat[eulerian2](n-k,n-k-j-1), j=(0..n-k-1))*(-1)^k fi end: for n from 0 to 9 do seq(A137375(n, k), k=(0..n/2)) od; # _Peter Luschny_, Dec 01 2012
%t A137375 Clear[p, x, t] p[t_] = Exp[x*(1 + t - Exp[t])]; Table[ ExpandAll[n!* SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a];
%t A137375 Table[Sum[Binomial[n, k - j] StirlingS2[n - k + j, j] (-1)^j, {j, 0, k}], {n, 0, 15}, {k, 0, n/2}] // Flatten (* _Eric W. Weisstein_, Nov 13 2018 *)
%o A137375 (Maxima) T(n,k):=sum(binomial(n,k-j)*stirling2(n-k+j,j)*(-1)^(j),j,0,k); /* _Vladimir Kruchinin_, Jan 13 2012 */
%o A137375 (Sage)
%o A137375 def A137375(n, k): return add(binomial(n,k-j)*(-1)^j*stirling_number2(n-k+j,j) for j in (0..k))
%o A137375 for n in range(11):
%o A137375     [A137375(n, k) for k in (0..n//2)]  # _Peter Luschny_, Dec 01 2012
%Y A137375 Row sums are: A000587.
%K A137375 tabf,sign
%O A137375 0,9
%A A137375 _Roger L. Bagula_, Apr 09 2008
%E A137375 Edited and simpler definition by _Peter Luschny_, Nov 27 2012