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A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

%I A129062 #68 Apr 18 2025 08:44:18
%S A129062 1,0,1,0,2,1,0,6,6,1,0,26,36,12,1,0,150,250,120,20,1,0,1082,2040,1230,
%T A129062 300,30,1,0,9366,19334,13650,4270,630,42,1,0,94586,209580,166376,
%U A129062 62160,11900,1176,56,1,0,1091670,2562354,2229444,952728,220500,28476,2016,72,1
%N A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.
%C A129062 Matrix product of Stirling2 with unsigned Stirling1 triangle.
%C A129062 For the subtriangle without column no. m=0 and row no. n=0 see A079641.
%C A129062 The reversed matrix product |S1|. S2 is given in A111596.
%C A129062 As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
%C A129062 E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
%C A129062 A048993*A132393 as infinite lower triangular matrices. - _Philippe Deléham_, Nov 01 2009
%C A129062 Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 19 2011.
%C A129062 Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%H A129062 Michael De Vlieger, <a href="/A129062/b129062.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A129062 Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
%H A129062 Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H A129062 Wolfdieter Lang, <a href="/A129062/a129062.txt">First ten rows and more</a>.
%F A129062 a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
%F A129062 E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
%F A129062 Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - _Philippe Deléham_, Nov 19 2011
%e A129062 Triangle begins:
%e A129062   1;
%e A129062   0,    1;
%e A129062   0,    2,    1;
%e A129062   0,    6,    6,    1;
%e A129062   0,   26,   36,   12,   1;
%e A129062   0,  150,  250,  120,  20,  1;
%e A129062   0, 1082, 2040, 1230, 300, 30,  1;
%p A129062 # The function BellMatrix is defined in A264428.
%p A129062 BellMatrix(n -> polylog(-n,1/2), 9); # _Peter Luschny_, Jan 27 2016
%t A129062 rows = 9;
%t A129062 t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
%t A129062 Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%t A129062 p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
%t A129062 Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* _Peter Luschny_, Jun 27 2019 *)
%o A129062 (Sage)
%o A129062 def a_row(n):
%o A129062     s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
%o A129062     return expand(s).list()
%o A129062 [a_row(n) for n in (0..9)] # _Peter Luschny_, Jun 28 2019
%Y A129062 Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.
%Y A129062 Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.
%K A129062 nonn,tabl,easy
%O A129062 0,5
%A A129062 _Wolfdieter Lang_, May 04 2007
%E A129062 New name by _Peter Luschny_, Jun 27 2019