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%I #32 Jun 22 2018 15:10:34
%S 1,3,1,9,9,1,27,63,18,1,81,405,225,30,1,243,2511,2430,585,45,1,729,
%T 15309,24381,9450,1260,63,1,2187,92583,234738,137781,28350,2394,84,1,
%U 6561,557685,2205225,1888110,563031,71442,4158,108,1
%N Stirling2 triangle with scaled diagonals (powers of 3).
%C This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
%C 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.
%C 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
%C Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H Andrew Howroyd, <a href="/A075498/b075498.txt">Table of n, a(n) for n = 1..1275</a>
%F a(n, m) = (3^(n-m)) * stirling2(n, m).
%F a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F 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.
%F G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
%F E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
%F From _Peter Bala_, Jan 13 2018: (Start)
%F Dobinski-type formulas for row polynomials R(n,x):
%F R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
%F R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
%F R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)
%e [1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
%e From _Philippe Deléham_, Feb 13 2013: (Start)
%e Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
%e 1;
%e 0, 1;
%e 0, 3, 1;
%e 0, 9, 9, 1;
%e 0, 27, 63, 18, 1;
%e 0, 81, 405, 225, 30, 1;
%e (End)
%p # The function BellMatrix is defined in A264428.
%p # Adds (1, 0, 0, 0, ..) as column 0.
%p BellMatrix(n -> 3^n, 9); # _Peter Luschny_, Jan 26 2016
%t Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%t rows = 9;
%t t = Table[3^n, {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o (PARI) for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.
%Y Cf. A075497, A075499.
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Oct 02 2002
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