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A059438
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Triangle T(n,k) (1<=k<=n) read by rows: T(n,k) = number of permutations of [1..n] with k components.
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12
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1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 32, 12, 4, 1, 461, 177, 58, 18, 5, 1, 3447, 1142, 327, 92, 25, 6, 1, 29093, 8411, 2109, 531, 135, 33, 7, 1, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).
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FORMULA
| Let f(x) = Sum_{n >= 0} n!*x^n, g(x) = 1-1/f(x). Then g(x) is g.f. for first diagonal A003319 and Sum_{n >= k} T(n, k)*x^n = g(x)^k.
Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007 where DELTA is Deleham's operator defined in A084938.
T(n+k, k)= Sum_{a_1 + a_2 +..+a_k = n} A003319(a_1)*A003319(a_2)*..*A003319(a_k). T(n, k) = 0 if n<k, T(n, 1)= A003319(n) and for n>=k T(n, k)= Sum_{j>=1} T(n-j, k-1)* A003319(j) . A059438 is formed from the self convolution of its first column (A003319). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 04 2004
Sum_{k>0} T(n, k) = n!; see A000142 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 05 2004
If g(x) = x+x^2+3*x^3+13*x^4+... is the generating function for the number of permutations with no global descents, then 1/(1-g(x)) is the generating function for n!. Setting t=1 in f(x, t) implies sum( T(n, k), k=1..n) = n!. Let g(x) be the o.g.f. for A003319. Then the o.g.f. for this table is given by f(x, t) = 1/(1-t*g(x))-1 (i.e. the coefficient of x^n*t^k in f(x, t) is T(n, k)). - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Jul 29 2004
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EXAMPLE
| 1; 1,1; 3,2,1; 13,7,3,1; ...
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CROSSREFS
| Diagonals give A003319, A059439, A059440, A055998.
Cf. A000007 A085771 A084938.
Sequence in context: A068440 A048647 A180190 * A156628 A104980 A134090
Adjacent sequences: A059435 A059436 A059437 * A059439 A059440 A059441
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Feb 01 2001
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 04 2001
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