%I #12 Mar 29 2017 02:59:22

%S 1,8,1,64,24,1,512,448,48,1,4096,7680,1600,80,1,32768,126976,46080,

%T 4160,120,1,262144,2064384,1232896,179200,8960,168,1,2097152,33292288,

%U 31653888,6967296,537600,17024,224,1

%N Stirling2 triangle with scaled diagonals (powers of 8).

%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(8*z) - 1)*x/8) - 1.

%H Andrew Howroyd, <a href="/A075503/b075503.txt">Table of n, a(n) for n = 1..1275</a>

%F a(n, m) = (8^(n-m)) * stirling2(n, m).

%F 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 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 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-8k*x), m >= 1.

%F E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.

%e [1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).

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

%e Triangle starts

%e * 1

%e * 8 1

%e * 64 24 1

%e * 512 448 48 1

%e * 4096 7680 1600 80 1

%e * 32768 126976 46080 4160 120 1

%e * 262144 2064384 1232896 179200 8960 168 1

%e * 2097152 33292288 31653888 6967296 537600 17024 224 1

%e (End)

%t Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)

%o (PARI) for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017

%Y Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.

%Y Cf. A075502, A075504.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Oct 02 2002