This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075498 #32 Jun 22 2018 15:10:34
%S A075498 1,3,1,9,9,1,27,63,18,1,81,405,225,30,1,243,2511,2430,585,45,1,729,
%T A075498 15309,24381,9450,1260,63,1,2187,92583,234738,137781,28350,2394,84,1,
%U A075498 6561,557685,2205225,1888110,563031,71442,4158,108,1
%N A075498 Stirling2 triangle with scaled diagonals (powers of 3).
%C A075498 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 A075498 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 A075498 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 A075498 Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H A075498 Andrew Howroyd, <a href="/A075498/b075498.txt">Table of n, a(n) for n = 1..1275</a>
%F A075498 a(n, m) = (3^(n-m)) * stirling2(n, m).
%F A075498 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 A075498 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 A075498 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
%F A075498 E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
%F A075498 From _Peter Bala_, Jan 13 2018: (Start)
%F A075498 Dobinski-type formulas for row polynomials R(n,x):
%F A075498 R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
%F A075498 R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
%F A075498 R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)
%e A075498 [1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
%e A075498 From _Philippe Deléham_, Feb 13 2013: (Start)
%e A075498 Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
%e A075498 1;
%e A075498 0, 1;
%e A075498 0, 3, 1;
%e A075498 0, 9, 9, 1;
%e A075498 0, 27, 63, 18, 1;
%e A075498 0, 81, 405, 225, 30, 1;
%e A075498 (End)
%p A075498 # The function BellMatrix is defined in A264428.
%p A075498 # Adds (1, 0, 0, 0, ..) as column 0.
%p A075498 BellMatrix(n -> 3^n, 9); # _Peter Luschny_, Jan 26 2016
%t A075498 Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%t A075498 rows = 9;
%t A075498 t = Table[3^n, {n, 0, rows}];
%t A075498 T[n_, k_] := BellY[n, k, t];
%t A075498 Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o A075498 (PARI) for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075498 Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.
%Y A075498 Cf. A075497, A075499.
%K A075498 nonn,easy,tabl
%O A075498 1,2
%A A075498 _Wolfdieter Lang_, Oct 02 2002