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A130534
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Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.
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29
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1, 1, 1, 2, 3, 1, 6, 11, 6, 1, 24, 50, 35, 10, 1, 120, 274, 225, 85, 15, 1, 720, 1764, 1624, 735, 175, 21, 1, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 362880, 1026576, 1172700, 723680, 269325, 63273
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| This triangle is an unsigned version of the triangle of Stirling numbers of the first kind, A008275, which is the main entry for these numbers. - N. J. A. Sloane, Jan 25 2011
Or, triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938 .
Reversal of A094638 .
Equals A132393*A007318, as infinite lower triangular matrices . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + .. ), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - .. ) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*stirling1(n,m), see the Maple program.
The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right hand columns) and A094587 (left hand columns).
See A163936 for information about the o.g.f.s. of the right hand columns of this triangle.
(End)
The number of elements greater than i to the left of i in a permutation gives the i-th element of the inversion vector. (Skiena-Pemmaraju 2003 p.69). T(n,k) is the number of n-permutations that have exactly k 0's in their inversion vector. See evidence in Mathematica code below. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 07 2010]
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REFERENCES
| Sriram Pemmaraju and Steven Skiena,Computational Discrete Mathematics,Cambridge University Press,2003,pp. 69-71 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 07 2010]
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009]
Dennis Walsh, A short note on unsigned Stirling numbers
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FORMULA
| T(0,0)=1, T(k,n)=0 if k>n or if n<0, T(n,k)=T(n-1,k-1)+n*T(n-1,k). T(n,0)=n!=A000142(n). T(2*n,n)=A129505(n+1). Sum_{k, 0<=k<=n}T(n,k)=(n+1)!=A000142(n+1). Sum_{k, 0<=k<=n}T(n,k)^2=A047796(n+1). T(n,k)=|Stirling1(n+1,k+1)|, see A008275 . (x+1)(x+2)...(x+n)=Sum_{k, 0<=k<=n}T(n,k)*x^k. [Corrected by Arie Bos, Jul 11 2008]
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007
For 1<=k<=n, let A={a1,a2,...,ak} denote a size-k subset of {1,2,...,n}. Then T(n,n-k)=sum(product(ai)) where the sum is over all subsets A and the product is over i=1,..,k. For example, T(4,1)=50 since (1)(2)(3)+(1)(2)(4)+(1)(3)(4)+(2)(3)(4)=50. Formula from Dennis P. Walsh, Jan 25 2011.
n-th row of the triangle = top row of M^n, where M is the production matrix:
1, 1
1, 2, 1
1, 3, 3, 1
1, 4, 6, 4, 1
...
- Gary W. Adamson, Jul 08 2011
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EXAMPLE
| Triangle begins:
1;
1, 1;
2, 3, 1;
6, 11, 6, 1;
24, 50, 35, 10, 1;
120, 274, 225, 85, 15, 1;
720, 1764, 1624, 735, 175, 21, 1;
5040, 13068, 13132, 6769, 1960, 322, 28, 1;
40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1;
362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1 ;...
T(3,2) = 6 because there are 6 permutations of {1,2,3,4} that have exactly 2 0's in their inversion vector:{1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {2, 1, 3, 4},{2, 3, 1, 4}, {2, 3, 4, 1}. The respective inversion vectors are: {0, 0, 1},{0, 1, 0}, {0, 2, 0}, {1, 0, 0}, {2, 0, 0}, {3, 0, 0}. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 07 2010]
T(3,1)=11 since there are exactly 11 permutations of {1,2,3,4} with exactly 2 cycles, namely, (1)(234),(1)(243),(2)(134),(2)(143),(3)(124),(3)(142),
(4)(123),(4)(143),(12)(34),(13)(24), and(14)(23).
[From Dennis P. Walsh, Jan 25 2011]
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
nmax:=10; with(combinat, stirling1): for n from 1 to nmax do for m from 1 to n do a(n, m):= (-1)^(n+m)*stirling1(n, m); T(n-1, m-1):= a(n, m): od: od: i:=0: for n from 0 to nmax-1 do for m from 0 to n do a(i):=T(n, m); i:=i+1: od: od: seq(a(n), n=0..i-1);
(End)
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MATHEMATICA
| Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 07 2010]
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CROSSREFS
| See A008275, which is the main entry for these numbers.
Diagonals : A000012 A000217 A000914 A001303 A000915 A053567 A112002. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
Row sums equal A000142.
The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.
Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.
(End)
Sequence in context: A121748 A174893 A008275 * A107416 A105613 A135894
Adjacent sequences: A130531 A130532 A130533 * A130535 A130536 A130537
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 09 2007
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