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A000914 Stirling numbers of the first kind: s(n+2, n).
(Formerly M1998 N0789)
43
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; text; internal format)
OFFSET

0,2

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, 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 Max Payton, Sep 06 2005

The sequence contains the partial sums of A006002, which represent the areas beneath lines created by the triangular numbers plotted (t(1),t(2)) connected to (t(2),t(3)) then (t(3),t(4))...(t(n-1),t(n)) and the x-axis. - J. M. Bergot, May 05 2012

a(n) = A052149(n+1)/2. - J. M. Bergot, Jun 02 2012

Number of functions f from [n+2] to [n+2] with f(x)=x for exactly n elements x of [n+2] and f(x)>x for exactly two elements x of [n+2]. To prove this, let the two elements of [n+2] with a larger image be labeled i and j. Note both i and j must be less than n+2. Then there are (n+2-i) choices for f(i) and (n+2-j) choices for f(j). Summing the product of the number of choices over all sets {i,j} gives us "Sum of product of unordered pairs of numbers from {1..n+1}" in the first line of the Comments Section. See the example in the Example Section below. - Dennis P. Walsh, Sep 06 2017

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).

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].

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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. - Simon Plouffe in his 1992 dissertation.

a(n) = Sum_{i=1..n} i*(i+1)^2/2. - Jon Perry, Jul 31 2003

-(3*n+2)*(n-1)*a(n) + (n+2)*(3*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 30 2015

a(n) = a(n-1) + (n+1)*binomial(n+1,2) for n >= 1. - Dennis P. Walsh, Sep 21 2015

a(n) = A001296(-2-n) for all n in Z. - Michael Somos, Sep 04 2017

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.

For n=2 the a(2)=11 functions f:[4]->[4] with exactly two f(x)=x and two f(x)>x are given by the 11 image vectors of form <f(1),f(2),f(3),f(4)> that follow: <1,3,4,4>, <1,4,4,4>, <2,2,4,4>, <3,2,4,4>, <4,2,4,4>, <2,3,3,4>, <2,4,3,4>, <3,3,3,4>, <3,4,3,4>, <4,3,3,4>, and <4,4,3,4>. - Dennis P. Walsh, Sep 06 2017

MAPLE

A000091 := n -> 1/24*(n+1)*n*(n+2)*(3*n+5);

A000091 := proc(n)

    combinat[stirling1](n+2, n) ;

end proc: # R. J. Mathar, May 19 2016

MATHEMATICA

s=0; lst={}; Do[s+=n^3-n^2; AppendTo[lst, s/2], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, May 25 2009 *)

Table[Sum[(i^3 - i^2)/2, {i, 0, n}], {n, 1, 41}] (* Zerinvary Lajos, Jul 10 2009 *)

Table[StirlingS1[n+2, n], {n, 0, 40}] (* Harvey P. Dale, Aug 24 2011 *)

a[ n_] := n (n + 1) (n + 2) (3 n + 5) / 24; (* Michael Somos, Sep 04 2017 *)

PROG

(PARI) a(n)=sum(i=1, n+1, sum(j=1, n+1, i*j*(i<j)))

(PARI) a(n)=sum(i=1, n+1, sum(j=1, i-1, i*j)) \\ Charles R Greathouse IV, Apr 07 2015

(PARI) a(n) = binomial(n+2, 3)*(3*n+5)/4 \\ Charles R Greathouse IV, Apr 07 2015

(Sage)[stirling_number1(n+2, n)for n in xrange(0, 38)]# Zerinvary Lajos, Mar 14 2009

(Haskell)

a000914 n = a000914_list !! n

a000914_list = scanl1 (+) a006002_list

-- Reinhard Zumkeller, Mar 25 2014

CROSSREFS

Cf. A000217, A000290, A033428, A033581, A033583, A008275.

Cf. similar sequences listed in A241765.

Cf. A001296.

Sequence in context: A204452 A041389 A205342 * A256317 A086735 A242300

Adjacent sequences:  A000911 A000912 A000913 * A000915 A000916 A000917

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000

Comments from Michael Somos, Jan 29 2000

Erroneous duplicate of the polynomial formula removed by R. J. Mathar, Sep 15 2009

STATUS

approved

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Last modified September 25 21:33 EDT 2017. Contains 292500 sequences.