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A000914
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Stirling numbers of first kind: s(n+2,n).
(Formerly M1998 N0789)
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23
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0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, 103385, 117800, 133672, 151096, 170170, 190995, 213675, 238317, 265031
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Sum of product of unordered pairs of numbers from {1..n+1}.
Number of edges of a complete k-partite graph of order k*(k+1)/2 (A000217), K_1,2,3,...,k. - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Oct 18 2001
This sequence holds the x^(n-2) coefficient of the characteristic polynomial of the N X N matrix A formed by MAX(i,j), where i is the row index and j is the column index of element A[i][j], 1 <= i,j <= N. Here N >= 2. - Paul M. Payton (paul.payton(AT)lmco.com), Sep 06 2005
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 4.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
H. S. Hall, S. R. Knight, Higher Algebra, Fourth Edition, Macmillan, 1891, p. 518.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n)=binomial(n+2, 3)*(3*n+5)/4 = (n+1)*n*(n+2)*(3*n+5)/24. E.g.f.: exp(x)*x*(48+84*x+32*x^2+3*x^3)/24. G.f.: (2*x+x^2)/(1-x)^5.
a(n)=sum(i=1, n, i*(i+1)^2/2) - Jon Perry (perry(AT)globalnet.co.uk), Jul 31 2003
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EXAMPLE
| Examples include, E(K_1,2,3)=s(2+2,2)=11 and E(K_1,2,3,4,5)=s(4+2,4)=85, where E is the function that counts edges of graphs.
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MAPLE
| f:= n -> 1/24*(n+1)*n*(n+2)*(3*n+5);
A000914:=-(2+z)/(z-1)**5; [S. Plouffe in his 1992 dissertation.]
a:=n->sum((n-j)^3-(n-j)^2, j=1..n): seq(a(n)/2, n=2..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 26 2008
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MATHEMATICA
| s=0; lst={}; Do[s+=n^3-n^2; AppendTo[lst, s/2], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 25 2009]
Table[Sum[(i^3 - i^2)/2, {i, 0, n}], {n, 1, 41}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
Table[StirlingS1[n+2, n], {n, 0, 40}] (* From Harvey P. Dale, Aug 24 2011 *)
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PROG
| (PARI) a(n)=sum(i=1, n+1, sum(j=1, n+1, i*j*(i<j)))
(Other) SAGE:[stirling_number1(n+2, n)for n in xrange(0, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]
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CROSSREFS
| Cf. A000217, A000290, A033428, A033581, A033583, A008275.
Sequence in context: A204452 A041389 A205342 * A086735 A078982 A154416
Adjacent sequences: A000911 A000912 A000913 * A000915 A000916 A000917
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000. Comments from Michael Somos, Jan 29, 2000.
Removed erroneous duplicate of the polynomial formula - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 15 2009
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