A000915 Stirling numbers of first kind s(n+4, n). (Formerly M5155 N2239)

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

1,1

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).

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

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 *)

Prog

(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

Crossrefs

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

Cf. A001298.

Sequence in context: A125412 A270846 A187048 * A006665 A010940 A045854

Adjacent sequences: A000912 A000913 A000914 * A000916 A000917 A000918

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved