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A075503 Stirling2 triangle with scaled diagonals (powers of 8).

%I A075503 #12 Mar 29 2017 02:59:22
%S A075503 1,8,1,64,24,1,512,448,48,1,4096,7680,1600,80,1,32768,126976,46080,
%T A075503 4160,120,1,262144,2064384,1232896,179200,8960,168,1,2097152,33292288,
%U A075503 31653888,6967296,537600,17024,224,1
%N A075503 Stirling2 triangle with scaled diagonals (powers of 8).
%C A075503 This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
%C A075503 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(8*z) - 1)*x/8) - 1.
%H A075503 Andrew Howroyd, <a href="/A075503/b075503.txt">Table of n, a(n) for n = 1..1275</a>
%F A075503 a(n, m) = (8^(n-m)) * stirling2(n, m).
%F A075503 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075503 a(n, m) = 8m*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 A075503 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-8k*x), m >= 1.
%F A075503 E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.
%e A075503 [1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
%e A075503 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075503 Triangle starts
%e A075503 *       1
%e A075503 *       8        1
%e A075503 *      64       24        1
%e A075503 *     512      448       48       1
%e A075503 *    4096     7680     1600      80      1
%e A075503 *   32768   126976    46080    4160    120     1
%e A075503 *  262144  2064384  1232896  179200   8960   168   1
%e A075503 * 2097152 33292288 31653888 6967296 537600 17024 224 1
%e A075503 (End)
%t A075503 Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%o A075503 (PARI) for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075503 Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.
%Y A075503 Cf. A075502, A075504.
%K A075503 nonn,easy,tabl
%O A075503 1,2
%A A075503 _Wolfdieter Lang_, Oct 02 2002