A001303 Stirling numbers of first kind, s(n+3, n), negated.

6, 50, 225, 735, 1960, 4536, 9450, 18150, 32670, 55770, 91091, 143325, 218400, 323680, 468180, 662796, 920550, 1256850, 1689765, 2240315, 2932776, 3795000, 4858750, 6160050, 7739550, 9642906, 11921175, 14631225, 17836160, 21605760, 26016936, 31154200

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is equal to the sum of the products of each distinct grouping of 3 members of the set {1, 2, 3, ..., n + 2} (a(1) = 1*2*3, a(2) = 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4, a(3) = 1*2*3 + 1*2*4 + 1*2*5 + 1*3*4 + 1*3*5 + 1*4*5 + 2*3*4 + 2*3*5 + 2*4*5 + 3*4*5). - Jeffreylee R. Snow, Sep 23 2013

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.
• Louis 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.
• 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 = 1..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]
• G. C. Greubel, A Note on Jain basis functions, arXiv:1612.09385 [math.CA], 2016.
• Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000217, A001297, A008275.

Programs

Maple

seq(numbperm (n,2)*numbperm (n,4)/48, n=4..33); # Zerinvary Lajos, Apr 26 2007
seq(15*binomial(n+2,6)-10*binomial(n+1,5)+binomial(n,4),n=4..30); # Miklos Kristof, Nov 04 2007
A001303 := proc(n)
    -combinat[stirling1](n+3,n) ;
end proc: # R. J. Mathar, May 19 2016

Mathematica

Table[-StirlingS1[n + 3, n], {n, 100}] (* T. D. Noe, Jun 27 2012 *)
a[ n_] := n (n + 1) (n + 2)^2 (n + 3)^2 / 48; (* Michael Somos, Sep 04 2017 *)

PARI

a(n) = n*(n+1)*(n+2)^2*(n+3)^2/48; \\ Altug Alkan, Aug 29 2017

Sage

[stirling_number1(n,n-3) for n in range(4, 34)] # Zerinvary Lajos, May 16 2009

Formula

a(n) = binomial(n+3, 4)*binomial(n+3, 2).

G.f.: x*(6 + 8*x + x^2)/(1 - x)^7. - Simon Plouffe in his 1992 dissertation

E.g.f. with offset 3: exp(x)*(6*(x^3)/3! + 26*(x^4)/4! + 35*(x^5)/5! + 15*(x^6)/6!). See row k=3 of A112486 for the coefficients [6, 26, 35, 15].

a(n) = (f(n+2, 3)/6!)*Sum_{m=0..min(3, n)} A112486(3,m)*f(6, 3-m)*f(n-1, m), with the falling factorials notation f(n, m):=n*(n-1)*...*(n-(m-1)).

From Jason Lang, Oct 03 2006: (Start)

a(n) = A000217(n) * n! / ( 4! * (n-4)! ) [for n > 4 and A000217 = the triangular numbers];

a(n) = ((n+4)! / n! ) ^2 / ( (n+2) * (n+1) * 2*4!);

a(n) = (n-0)^2 * (n-1)^2 * (n-2) * (n-3) / (2*4!). (End)

From Miklos Kristof, Nov 04 2007: (Start)

a(n) = 15*binomial(n+5,6) - 10*binomial(n+4,5) + binomial(n+3,4).

E.g.f. with offset 4: exp(x)*((1/4)*x^4 + (1/6)*x^5 + (1/48)*x^6). (End)

a(n) = n*(n+1)(n+2)^2*(n+3)^2/48. - Jeremy Galvagni, Mar 03 2009

From Gary Detlefs, Jun 06 2010: (Start)

a(n) = (n+3)^2/(n^2-1)*a(n-1), n > 1;

a(n) = 6*Product_{k=2..n} (k+3)^2/(k^2 - 1). (End)

a(n) = A001297(-3-n) for all n in Z. - Michael Somos, Sep 04 2017

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 472/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*Pi^2/3 + 16/9 - 64*log(2)/3. (End)

Extensions

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

Notation of the polynomial formula edited by R. J. Mathar, Sep 15 2009