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A089231
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Triangular array A066667 or A008297 unsigned and transposed.
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10
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1, 1, 2, 1, 6, 6, 1, 12, 36, 24, 1, 20, 120, 240, 120, 1, 30, 300, 1200, 1800, 720, 1, 42, 630, 4200, 12600, 15120, 5040, 1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880
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OFFSET
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1,3
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COMMENTS
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T(n, k) is also the number of nilpotent partial one-one bijections (of an n-element set) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Sep 14 2008
T(n, k) is also the number of acyclic directed graphs on n labeled nodes with k-1 edges with all indegrees and outdegrees at most 1. - Felix A. Pahl, Dec 25 2012
For n > 1, the n-th derivative of exp(1/x) is of the form (exp(1/x)/x^(2*n))*(P(n-1,x)) where P(n-1,x) is a polynomial of degree n-1 with n terms. The term of degree k in P(n-1,x) has a coefficient given by T(n-1,k). Example: The third derivative of exp(1/x) is (exp(1/x)/x^6)*(1+6x+6x^2) and the 3rd row of this triangle is 1, 6, 6, which corresponds to this coefficients of the polynomial 1+6x+6x^2. - Derek Orr, Nov 06 2014
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 203.
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LINKS
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FORMULA
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T(n, k) = C(n, n-k+1)*(n-1)!/(n-k)! = Sum_{i=n-k+1..n} |S1(n, i)*S2(i, n-k+1)| , with S1, S2 the Stirling numbers.
Each row represents a polynomial:
P(1,x) = 1;
P(2,x) = 1 + 2x;
P(3,x) = 1 + 6x + 6x^2;
P(4,x) = 1 + 12x + 36x^2 + 24x^3;
...
They are related through P(n+1,x) = x^2*P'(n,x) - (1+2*n*x)*P(n,x) with P(1,x) = 1.
(End)
Working with an offset of 0:
G.f.: exp(x*t)*I_1(2*sqrt(x)) = 1 + (1 + 2*t)*x/(1!*2!) + (1 + 6*t + 6*t^2)*x^2/(2!*3!) + (1 + 12*t + 36*t^2 + 24*t^3)*x^3/(3!*4!) + ..., where I_1(x) = Sum_{n >= 0} (x/2)^(2*n)/(n!*(n+1)!) is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*(n+1)!/(k+1)!* binomial(n,k)*t^(n-k)*R(k,t). Cf. A144084. (End)
The following formulas use a column index k starting at 0:
E.g.f.: exp(x/(1 - t*x)) - 1 = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 6*t^2)*x^3/3! + ....
Recurrence for row polynomials: R(n+1,t) = (1 + 2*n*t)R(n,t) - n*(n-1)*t^2*R(n-1,t), with R(1,t) = 1 and R(2,t) = 1 + 2*t.
R(n+1,t) equals the numerator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The denominator polynomial is the n-th row polynomial of A144084. (End)
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EXAMPLE
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1;
1, 2;
1, 6, 6;
1, 12, 36, 24;
1, 20, 120, 240, 120;
1, 30, 300, 1200, 1800, 720;
1, 42, 630, 4200, 12600, 15120, 5040;
1, 56, 1176, 11760, 58800, 141120, 141120, 40320;
1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880;
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MAPLE
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P := n -> simplify(hypergeom([-n, -n+1], [], 1/t));
seq(print(seq(coeff(expand(t^k*P(k)), t, k-j+1), j=1..k)), k=1..n); # Peter Luschny, Oct 29 2014
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MATHEMATICA
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Table[(Binomial[n - 1, k - 1] Binomial[n, k - 1]/k) k!, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jul 04 2016 *)
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PROG
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(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n, k)/(n-k+1)!, ", "); ); print(); ); } \\ Michel Marcus, Jan 12 2016
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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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