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%I M1998 N0789 #133 Sep 08 2022 08:44:28
%S 0,2,11,35,85,175,322,546,870,1320,1925,2717,3731,5005,6580,8500,
%T 10812,13566,16815,20615,25025,30107,35926,42550,50050,58500,67977,
%U 78561,90335,103385,117800,133672,151096,170170,190995,213675,238317,265031
%N Stirling numbers of the first kind: s(n+2, n).
%C Sum of product of unordered pairs of numbers from {1..n+1}.
%C 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
%C 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
%C 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
%C 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
%D 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.
%D George E. Andrews, Number Theory, Dover Publications, New York, 1971, p. 4.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D H. S. Hall and S. R. Knight, Higher Algebra, Fourth Edition, Macmillan, 1891, p. 518.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000914/b000914.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Karl Dienger, <a href="/A000217/a000217.pdf">Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung</a>, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(n+2, 3)*(3*n+5)/4 = (n+1)*n*(n+2)*(3*n+5)/24.
%F E.g.f.: exp(x)*x*(48 + 84*x + 32*x^2 + 3*x^3)/24.
%F G.f.: (2*x+x^2)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) = Sum_{i=1..n} i*(i+1)^2/2. - _Jon Perry_, Jul 31 2003
%F a(n) = A052149(n+1)/2. - _J. M. Bergot_, Jun 02 2012
%F -(3*n+2)*(n-1)*a(n) + (n+2)*(3*n+5)*a(n-1) = 0. - _R. J. Mathar_, Apr 30 2015
%F a(n) = a(n-1) + (n+1)*binomial(n+1,2) for n >= 1. - _Dennis P. Walsh_, Sep 21 2015
%F a(n) = A001296(-2-n) for all n in Z. - _Michael Somos_, Sep 04 2017
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 162*log(3)/5 - 18*sqrt(3)*Pi/5 - 384/25.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 36*sqrt(3)*Pi/5 - 96*log(2)/5 - 636/25. (End)
%e 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.
%e 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
%p A000914 := n -> 1/24*(n+1)*n*(n+2)*(3*n+5);
%p A000914 := proc(n)
%p combinat[stirling1](n+2,n) ;
%p end proc: # _R. J. Mathar_, May 19 2016
%t Table[StirlingS1[n+2,n],{n,0,40}] (* _Harvey P. Dale_, Aug 24 2011 *)
%t a[ n_] := n (n + 1) (n + 2) (3 n + 5) / 24; (* _Michael Somos_, Sep 04 2017 *)
%o (PARI) a(n)=sum(i=1,n+1,sum(j=1,n+1,i*j*(i<j)))
%o (PARI) a(n)=sum(i=1,n+1,sum(j=1,i-1,i*j)) \\ _Charles R Greathouse IV_, Apr 07 2015
%o (PARI) a(n) = binomial(n+2, 3)*(3*n+5)/4 \\ _Charles R Greathouse IV_, Apr 07 2015
%o (Sage) [stirling_number1(n+2, n) for n in range(41)] # _Zerinvary Lajos_, Mar 14 2009
%o (Haskell)
%o a000914 n = a000914_list !! n
%o a000914_list = scanl1 (+) a006002_list
%o -- _Reinhard Zumkeller_, Mar 25 2014
%o (Magma) [StirlingFirst(n+2, n): n in [0..40]]; // _Vincenzo Librandi_, May 28 2019
%Y Cf. A000217, A000290, A033428, A033581, A033583, A008275, A052149.
%Y Cf. similar sequences listed in A241765.
%Y Cf. A001296.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
%E Comments from _Michael Somos_, Jan 29 2000
%E Erroneous duplicate of the polynomial formula removed by _R. J. Mathar_, Sep 15 2009
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