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Stirling2 triangle with scaled diagonals (powers of 6).
9

%I #20 Jun 23 2018 07:25:56

%S 1,6,1,36,18,1,216,252,36,1,1296,3240,900,60,1,7776,40176,19440,2340,

%T 90,1,46656,489888,390096,75600,5040,126,1,279936,5925312,7511616,

%U 2204496,226800,9576,168,1,1679616,71383680

%N Stirling2 triangle with scaled diagonals (powers of 6).

%C This is a lower triangular infinite matrix of the Jabotinsky type. See the 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(6*z) - 1)*x/6) - 1.

%H Andrew Howroyd, <a href="/A075501/b075501.txt">Table of n, a(n) for n = 1..1275</a>

%F a(n, m) = (6^(n-m)) * stirling2(n, m).

%F a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.

%F a(n, m) = 6m*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}(1-6k*x), m >= 1.

%F E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.

%e [1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).

%e From _Andrew Howroyd_, Mar 25 2017: (Start)

%e Triangle starts

%e * 1

%e * 6 1

%e * 36 18 1

%e * 216 252 36 1

%e * 1296 3240 900 60 1

%e * 7776 40176 19440 2340 90 1

%e * 46656 489888 390096 75600 5040 126 1

%e * 279936 5925312 7511616 2204496 226800 9576 168 1

%e (End)

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> 6^n, 9); # _Peter Luschny_, Jan 28 2016

%t Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 10;

%t M = BellMatrix[6^#&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)

%o (PARI) for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017

%Y Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.

%Y Cf. A075500, A075502.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Oct 02 2002