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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. 10
 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, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Matrix product of Stirling2 with unsigned Stirling1 triangle. For the subtriangle without column nr. m=0 and row nr. n=0 see A079641. The reversed matrix product |S1|. S2 is given in A111596. 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. 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. A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009 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. Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened). Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020. Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020. W. Lang, First ten rows and more. FORMULA a(n,m) = sum(S2(n,k)*|S1(k,m)|, k=m..n), n>=0; S2=A048993, S1=A048994. 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)). 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 EXAMPLE Triangle begins: 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, 300, 30,  1; MAPLE # The function BellMatrix is defined in A264428. BellMatrix(n -> polylog(-n, 1/2), 9); # Peter Luschny, Jan 27 2016 MATHEMATICA rows = 9; t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}]; Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *) PROG (Sage) def a_row(n):     s = sum(stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))     return expand(s).list() [a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019 CROSSREFS Cf. A000629, A000670, A005649, A079641, A325872, A325873. Sequence in context: A111596 A271703 A276922 * A281662 A163936 A288874 Adjacent sequences:  A129059 A129060 A129061 * A129063 A129064 A129065 KEYWORD nonn,tabl,easy AUTHOR Wolfdieter Lang, May 04 2007 EXTENSIONS New name by Peter Luschny, Jun 27 2019 STATUS approved

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Last modified January 20 10:55 EST 2022. Contains 350472 sequences. (Running on oeis4.)