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 A179455 Triangle read by rows: number of permutation trees of power n and height <= k + 1. 6
 1, 1, 1, 2, 1, 5, 6, 1, 15, 23, 24, 1, 52, 106, 119, 120, 1, 203, 568, 700, 719, 720, 1, 877, 3459, 4748, 5013, 5039, 5040, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Partial row sums of A179454. Special cases: A179455(n,1) = BellNumber(n) = A000110(n) for n > 1; A179455(n,n-1) = n! for n > 1 and A179455(n,n-2) = A033312(n) for n > 1. Column 3 is A187761(n) for n >= 3. See the interpretation of Joerg Arndt in A187761: Maps such that f^[k](x) = f^[k-1](x) correspond to column k of A179455 (for n >= k). - Peter Luschny, Jan 08 2013 LINKS Alois P. Heinz, Rows n = 0..141, flattened Peter Luschny, Permutation Trees EXAMPLE As a (0,0)-based triangle with an additional column [1,0,0,0,...] at the left hand side: 1; 0, 1; 0, 1, 2; 0, 1, 5, 6; 0, 1, 15, 23, 24; 0, 1, 52, 106, 119, 120; 0, 1, 203, 568, 700, 719, 720; 0, 1, 877, 3459, 4748, 5013, 5039, 5040; 0, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320; 0, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880; MATHEMATICA b[n_, t_, h_] := b[n, t, h] = If[n == 0 || h == 0, 1, Sum[Binomial[n - 1, j - 1]*b[j - 1, 0, h - 1]*b[n - j, t, h], {j, 1, n}]]; T[0, 0] = 1; T[n_, k_] := b[n, 1, k]; Table[T[n, k], {n, 0, 9}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, Jul 10 2019, after Alois P. Heinz in A179454 *) PROG (Sage) # Generating algorithm from Joerg Arndt. def A179455row(n):     def generate(n, k):         if n == 0 or k == 0: return 0         for j in range(n-1, 0, -1):             f = a[j] + 1             while f <= j:                 a[j] = f1 = fl = f                 for i in range(k):                     fl = f1                     f1 = a[fl]                 if f1 == fl: return j                 f += 1             a[j] = 0         return 0     count = [1 for j in range(n)] if n > 0 else [1]     for k in range(n):         a = [0 for j in range(n)]         while generate(n, k) != 0:             count[k] += 1     return count for n in range(9): A179455row(n) # Peter Luschny, Jan 08 2013 (Sage) # Alternatively, based on the function bell_transform defined in A264428: # Adds the column (1, 0, 0, 0, ..) to the left hand side and starts at n=0. def A179455_matrix(dim):     b = [1]+[0]*(dim-1); L = [b]     for k in range(dim):         b = [sum(bell_transform(n, b)) for n in range(dim)]         L.append(b)     return matrix(ZZ, dim, lambda n, k: L[k][n] if k<=n else 0) print A179455_matrix(10) # Peter Luschny, Dec 06 2015 CROSSREFS Row sums are A264151. Cf. A000110, A179454, A179456, A187761, A264428. Sequence in context: A095801 A128567 A217204 * A039810 A328297 A124575 Adjacent sequences:  A179452 A179453 A179454 * A179456 A179457 A179458 KEYWORD nonn,tabf,look,nice AUTHOR Peter Luschny, Aug 11 2010 STATUS approved

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Last modified October 14 15:02 EDT 2019. Contains 328019 sequences. (Running on oeis4.)