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A245049
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Number A(n,k) of hybrid k-ary trees with n internal nodes; square array A(n,k), n>=0, k>=1, read by antidiagonals.
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11
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1, 1, 2, 1, 2, 3, 1, 2, 7, 5, 1, 2, 11, 31, 8, 1, 2, 15, 81, 154, 13, 1, 2, 19, 155, 684, 820, 21, 1, 2, 23, 253, 1854, 6257, 4575, 34, 1, 2, 27, 375, 3920, 24124, 60325, 26398, 55, 1, 2, 31, 521, 7138, 66221, 331575, 603641, 156233, 89, 1, 2, 35, 691, 11764, 148348, 1183077, 4736345, 6210059, 943174, 144
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OFFSET
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0,3
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LINKS
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FORMULA
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A(n,k) = 1/((k-1)*n+1) * Sum_{i=0..n} C((k-1)*n+i,i)*C((k-1)*n+i+1,n-i).
A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).
G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, ...
3, 7, 11, 15, 19, 23, 27, ...
5, 31, 81, 155, 253, 375, 521, ...
8, 154, 684, 1854, 3920, 7138, 11764, ...
13, 820, 6257, 24124, 66221, 148348, 290305, ...
21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...
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MAPLE
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A:= (n, k)-> add(binomial((k-1)*n+i, i)*
binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);
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MATHEMATICA
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A[n_, k_] := Sum[Binomial[(k-1)*n+i, i]*Binomial[(k-1)*n+i+1, n-i], {i, 0, n}]/((k-1)*n+1); Table[A[n, 1+d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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