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 A075498 Stirling2 triangle with scaled diagonals (powers of 3). 14
 1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1 (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 D. E. 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(3*z) - 1)*x/3) - 1. Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013 Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA a(n, m) = (3^(n-m)) * stirling2(n, m). a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0. a(n, m) = 3*m*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}(1-3*k*x), m >= 1. E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1. From Peter Bala, Jan 13 2018: (Start) Dobinski-type formulas for row polynomials R(n,x): R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!; R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!. R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End) EXAMPLE [1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2). From Philippe Deléham, Feb 13 2013: (Start) Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:   1;   0,   1;   0,   3,   1;   0,   9,   9,   1;   0,  27,  63,  18,   1;   0,  81, 405, 225,  30,   1; (End) MAPLE # The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016 MATHEMATICA Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *) rows = 9; t = Table[3^n, {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) PROG (PARI) for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017 CROSSREFS Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212. Cf. A075497, A075499. Sequence in context: A078416 A223533 A021973 * A105729 A104750 A163394 Adjacent sequences:  A075495 A075496 A075497 * A075499 A075500 A075501 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Oct 02 2002 STATUS approved

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Last modified April 1 04:55 EDT 2020. Contains 333155 sequences. (Running on oeis4.)