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%I #53 Apr 17 2021 01:39:19
%S 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 300,30,1,0,9366,19334,13650,4270,630,42,1,0,94586,209580,166376,
%U 62160,11900,1176,56,1,0,1091670,2562354,2229444,952728,220500,28476,2016,72,1
%N 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 Matrix product of Stirling2 with unsigned Stirling1 triangle.
%C For the subtriangle without column nr. m=0 and row nr. n=0 see A079641.
%C The reversed matrix product |S1|. S2 is given in A111596.
%C 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 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 A048993*A132393 as infinite lower triangular matrices. - _Philippe Deléham_, Nov 01 2009
%C 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 Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%H Michael De Vlieger, <a href="/A129062/b129062.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H 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 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 W. Lang, <a href="/A129062/a129062.txt">First ten rows and more.</a>
%F a(n,m) = sum(S2(n,k)*|S1(k,m)|, k=m..n), n>=0; S2=A048993, S1=A048994.
%F E.g.f. column nr. m (with leading zeros): (f(x)^m)/m! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
%F 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 Triangle begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 6, 6, 1;
%e 0, 26, 36, 12, 1;
%e 0, 150, 250, 120, 20, 1;
%e 0, 1082, 2040, 1230, 300, 30, 1;
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> polylog(-n,1/2), 9); # _Peter Luschny_, Jan 27 2016
%t rows = 9;
%t t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%t p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
%t Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* _Peter Luschny_, Jun 27 2019 *)
%o (Sage)
%o def a_row(n):
%o s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
%o return expand(s).list()
%o [a_row(n) for n in (0..9)] # _Peter Luschny_, Jun 28 2019
%Y Cf. A000629, A000670, A005649, A079641, A325872, A325873.
%K nonn,tabl,easy
%O 0,5
%A _Wolfdieter Lang_, May 04 2007
%E New name by _Peter Luschny_, Jun 27 2019
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