A001303 Stirling numbers of first kind, s(n+3, n), negated. (Formerly M4258 N1779)

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

1,1

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).

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)

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 *)

Prog

(Sage) [stirling_number1(n, n-3) for n in range(4, 34)] # Zerinvary Lajos, May 16 2009
(PARI) a(n) = n*(n+1)*(n+2)^2*(n+3)^2/48; \\ Altug Alkan, Aug 29 2017

Crossrefs

Cf. A000217, A001297, A008275.

Sequence in context: A035290 A138422 A356251 * A220887 A213807 A241781

Adjacent sequences: A001300 A001301 A001302 * A001304 A001305 A001306

Keyword

nonn,easy

Author

N. J. A. Sloane

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

Status

approved