A001298 Stirling numbers of the second kind S(n+4, n). (Formerly M5222 N2272)

0, 1, 31, 301, 1701, 6951, 22827, 63987, 159027, 359502, 752752, 1479478, 2757118, 4910178, 8408778, 13916778, 22350954, 34952799, 53374629, 79781779, 116972779, 168519505, 238929405, 333832005, 460192005, 626551380, 843303006, 1122998436, 1480692556

Offset

0,3

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. 835.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

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 = 0..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].

M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.

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

Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Formula

G.f.: x(1 + 22x + 58x^2 + 24x^3)/(1 - x)^9. - Paul Barry, Aug 05 2004

a(n) = Stirling2(n+4, n) = Sum_{L=1..n} (Sum_{k=1..L} (Sum_{j=1..k} (Sum_{i=1..j} i*j*k*L))) = (n+4)*(n+3)*(n+2)*(n+1)*n *(15*n^3 + 30*n^2 + 5*n - 2)/5760 = (15*n^3 + 30*n^2 + 5*n - 2)*binomial(n+4, 5)/48. - Vladeta Jovovic, Jan 31 2005

E.g.f. with offset -3: exp(x)*(1*(x^4)/4! + 26*(x^5)/5! + 130*(x^6)/6! + 210*(x^7)/7! +105*(x^8)/8!). For the coefficients [1, 26, 130, 210, 105] see triangle A112493. E.g.f.: x*exp(x)*(15*x^7 + 600*x^6 + 8600*x^5 + 55248*x^4 + 162960*x^3 + 202560*x^2 + 83520*x + 5760)/5760. Above given e.g.f. differentiated three times.

O.g.f. is D^4(x/(1-x)), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012

a(n) = A000915(-4-n) for all n in Z. - Michael Somos, Sep 04 2017

Maple

A001298:=-(1+22*z+58*z**2+24*z**3)/(z-1)**9; # Simon Plouffe in his 1992 dissertation, without the leading 0

Mathematica

Table[StirlingS2[n+4, n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
a[ n_] := n (n + 1) (n + 2) (n + 3) (n + 4) (15 n^3 + 30 n^2 + 5 n - 2) / 5760; (* Michael Somos, Sep 04 2017 *)

Prog

(PARI) {a(n) = n * (n+1) * (n+2) * (n+3) * (n+4) * (15*n^3 + 30*n^2 + 5*n - 2) / 5760}; /* Michael Somos, Sep 04 2017 */
(Sage) [stirling_number2(n+4, n) for n in range(0, 24)] # Zerinvary Lajos, May 16 2009
(Magma) [n*(n+1)*(n+2)*(n+3)*(n+4)*(15*n^3 + 30*n^2 + 5*n - 2)/5760: n in [0..50]]; // G. C. Greubel, Oct 22 2017

Crossrefs

Cf. A001296, A001297, A008277, A008517, A094262.

Cf. A000915.

Sequence in context: A327561 A279980 A369173 * A027841 A221853 A000500

Adjacent sequences: A001295 A001296 A001297 * A001299 A001300 A001301

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Name edited and initial zero added by Nathaniel Johnston, Apr 30 2011

Status

approved