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 A075503 Stirling2 triangle with scaled diagonals (powers of 8). 9
 1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays. 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA a(n, m) = (8^(n-m)) * stirling2(n, m). a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0. 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. G.f. for m-th column: (x^m)/Product_{k=1..m}(1-8k*x), m >= 1. E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1. EXAMPLE [1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2). From Andrew Howroyd, Mar 25 2017: (Start) Triangle starts *       1 *       8        1 *      64       24        1 *     512      448       48       1 *    4096     7680     1600      80      1 *   32768   126976    46080    4160    120     1 *  262144  2064384  1232896  179200   8960   168   1 * 2097152 33292288 31653888 6967296 537600 17024 224 1 (End) MATHEMATICA Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *) PROG (PARI) for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017 CROSSREFS Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507. Cf. A075502, A075504. Sequence in context: A089276 A051932 A038279 * A260040 A051379 A143499 Adjacent sequences:  A075500 A075501 A075502 * A075504 A075505 A075506 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Oct 02 2002 STATUS approved

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Last modified September 26 11:29 EDT 2020. Contains 337367 sequences. (Running on oeis4.)