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 A075500 Stirling2 triangle with scaled diagonals (powers of 5). 10
 1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays. The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA a(n, m) = (5^(n-m)) * stirling2(n, m). a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0. a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1. G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1. E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1. EXAMPLE [1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2). From Andrew Howroyd, Mar 25 2017: (Start) Triangle starts *     1 *     5       1 *    25      15       1 *   125     175      30       1 *   625    1875     625      50      1 *  3125   19375   11250    1625     75    1 * 15625  196875  188125   43750   3500  105   1 * 78125 1984375 3018750 1063125 131250 6650 140 1 (End) MAPLE # The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016 MATHEMATICA Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *) BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]]; rows = 10; M = BellMatrix[5^#&, rows]; Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *) PROG (PARI) for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017 CROSSREFS Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1). Cf. A075499, A075501. Sequence in context: A193685 A174358 A264131 * A096645 A140713 A125906 Adjacent sequences:  A075497 A075498 A075499 * A075501 A075502 A075503 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Oct 02 2002 STATUS approved

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Last modified February 21 23:21 EST 2020. Contains 332113 sequences. (Running on oeis4.)