A000915 Stirling numbers of first kind s(n+4, n).

24, 274, 1624, 6769, 22449, 63273, 157773, 357423, 749463, 1474473, 2749747, 4899622, 8394022, 13896582, 22323822, 34916946, 53327946, 79721796, 116896626, 168423871, 238810495, 333685495, 460012995, 626334345, 843041745, 1122686019, 1480321269, 1933889244

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• 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.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 259.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 48.
• 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].
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A008275, A094216, A001303 for s(n+3,n), A053567 for s(n+5,n).

Cf. A001298.

Programs

Maple

A000915 := proc(n)
    combinat[stirling1](n+4,n) ;
end proc:
seq(A000915(n),n=1..10) ; # R. J. Mathar, May 19 2016

Mathematica

Table[Binomial[n + 4, 5]*(15*n^3 + 150*n^2 + 485*n + 502)/48, {n, 50}] (* T. D. Noe, Jun 20 2012 *)
a[ n_] := n (n + 1) (n + 2) (n + 3) (n + 4) (15 n^3 + 150 n^2 + 485 n + 502) / 5760; (* Michael Somos, Sep 04 2017 *)

PARI

{a(n) = n * (n+1) * (n+2) * (n+3) * (n+4) * (15*n^3+ 150*n^2 + 485*n + 502) / 5760}; /* Michael Somos, Sep 04 2017 */

Sage

[stirling_number1(n,n-4) for n in range(5, 30)] # Zerinvary Lajos, May 16 2009

Formula

a(n) = binomial(n+4, 5)*(15*n^3 + 150*n^2 + 485*n + 502)/48. - André F. Labossière, Sep 30 2004

Stirling1(n+1, n-3) = Sum_{L=1..n} (Sum_{k=L+1..n} (Sum_{j=k+1..n} (Sum_{i=j+1..n} i*j*k*L))), cf. A001298. - Vladeta Jovovic, Jan 31 2005

E.g.f. with offset 4: exp(x)*(Sum_{m=0..4} A112486(4,m)*(x^(4+m))/(4+m)!).

a(n) = (f(n+3, 4)/8!)*Sum_{m=0..min(4, n-1)} A112486(4,m)*f(8, 4-m)*f(n-1, m), with the falling factorials f(n, m):=n*(n-1)*...*(n-(m-1)).

G.f.: x*(24 + 58*x + 22*x^2 + x^3)/(1 - x)^9, see the k=3 row of triangle A112007 for [24, 58, 22, 1].

a(n) = A001298(-4-n) for all n in Z. - Michael Somos, Sep 04 2017

Extensions

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