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 A090238 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. 11
 1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 16, 6, 1, 0, 120, 72, 30, 8, 1, 0, 720, 372, 152, 48, 10, 1, 0, 5040, 2208, 828, 272, 70, 12, 1, 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1, 0, 362880, 115200, 33192, 9696, 2720, 664, 126, 16, 1, 0, 3628800, 996480, 247968, 64704, 17312, 4380, 952, 160, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - David Callan, Nov 29 2007 For n > 0, -Sum_{i=0..n} (-1)^i*T(n,i) is the number of indecomposable permutations A003319. - Peter Luschny, Mar 13 2009 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34. LINKS FORMULA T(n, k) = T(n-1, k-1) + ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n > 0, T(0, k)=0 if k > 0. G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n. Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m). - Philippe Deléham, Jan 31 2004 T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - Philippe Deléham, Feb 18 2004 From Peter Bala, May 27 2017: (Start) Conjectural o.g.f.: 1/(1 + t - t*F(x)) = 1 + t*x + (2*t + t^2)*x^2 + (6*t + 4*t^2 + t^3)*x^3 + ..., where F(x) = Sum_{n >= 0} n!*x^n. If true then a continued fraction representation of the o.g.f. is 1 - t + t/(1 - x/(1 - t*x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End) EXAMPLE Triangle begins:   1;   0,       1;   0,       2,      1;   0,       6,      4,      1;   0,      24,     16,      6,     1;   0,     120,     72,     30,     8,     1;   0,     720,    372,    152,    48,    12,     1;   0,    5040,   2208,    828,   272,    70,    12,    1;   0,   40320,  14976,   4968,  1576,   440,    96,   14,   1;   0,  366880, 115200,  33192,  9696,  2720,   664,  126,  16,   1;   0, 3628800, 996480, 247968, 64704, 17312,  4380,  952, 160,  18,  1;   ... MAPLE T := proc(n, k) option remember; if n=0 and k=0 then return 1 fi; if n>0 and k=0 or k>0 and n=0 then return 0 fi; T(n-1, k-1)+(n+k-1)*T(n-1, k)/k end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # Peter Luschny, Mar 03 2016 MATHEMATICA T[n_, k_] := T[n, k] = T[n-1, k-1] + ((n+k-1)/k)*T[n-1, k]; T[0, 0] = 1; T[_, 0] = T[0, _] = 0; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018 *) CROSSREFS Another version: A059369. Row sums: A051296, A003319 (n>0). Diagonals: A000007, A000142, A059371, A000012, A005843, A054000. Sequence in context: A205813 A127631 A122538 * A047922 A276891 A021830 Adjacent sequences:  A090235 A090236 A090237 * A090239 A090240 A090241 KEYWORD easy,nonn,tabl AUTHOR Philippe Deléham, Jan 23 2004, Jun 14 2007 STATUS approved

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Last modified January 27 13:04 EST 2022. Contains 350607 sequences. (Running on oeis4.)