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A038455
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Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).
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5
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1, 3, 1, 20, 9, 1, 210, 107, 18, 1, 3024, 1650, 335, 30, 1, 55440, 31594, 7155, 805, 45, 1, 1235520, 725592, 176554, 22785, 1645, 63, 1, 32432400, 19471500, 4985316, 705649, 59640, 3010, 84, 1, 980179200, 598482000, 159168428, 24083892, 2267769, 136080, 5082, 108, 1
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OFFSET
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1,2
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COMMENTS
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Original name: A Jabotinsky-triangle related to A006963.
i) This triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z) = c(z) with c(z) the g.f. for the Catalan numbers A000108. (Notation of F(z) as in Knuth's paper).
ii) E(n, x) = Sum_{m=1..n} a(n, m)*x^m, E(0, x) = 1, are exponential convolution polynomials: E(n, x + y) = Sum_{k=0..n} binomial(n, k)*E(k, x)*E(n-k, y), (cf. Knuth's paper with E(n, x)= n!*F(n, x).)
iii) Explicit formula: see Knuth's paper for f(n, m) formula with f(k) = A006963(k + 1).
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LINKS
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FORMULA
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a(n, 1) = A006963(n + 1) = (2*n - 1)!/n!, n >= 1;
a(n, m) = Sum_{j=1..n-m+1} binomial(n - 1, j - 1)*A006963(j + 1)*a(n - j, m - 1), n >= m >= 2.
a(n, m) = (n - 1)!*(Sum_{k=m..n} Stirling1(k, m)*binomial(2*n, n-k)/(k-1)!). - Vladimir Kruchinin, Mar 26 2013
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EXAMPLE
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Triangle starts:
[1] 1;
[2] 3, 1;
[3] 20, 9, 1;
[4] 210, 107, 18, 1;
[5] 3024, 1650, 335, 30, 1;
[6] 55440, 31594, 7155, 805, 45, 1;
[7] 1235520, 725592, 176554, 22785, 1645, 63, 1;
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MAPLE
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# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> (2*n+1)!/(n+1)!, 9); # Peter Luschny, Jan 28 2016
gf := n -> x*pochhammer(n + x, n)/(n + x):
ser := n -> series(gf(n), x, n + 2):
seq(seq(coeff(ser(n), x, k), k = 1..n), n = 1..9); # Peter Luschny, Jun 27 2024
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MATHEMATICA
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BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11; M = BellMatrix[(2#+1)!/(#+1)!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten
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PROG
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(Maxima)
a(n, m):=(n-1)!*(sum((stirling1(k, m)*binomial(2*n, n-k))/(k-1)!, k, m, n)); /* Vladimir Kruchinin, Mar 26 2013 */
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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