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

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

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. 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.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
• 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).

Crossrefs

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

Cf. A000915.

Programs

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

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 *)

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

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

Extensions

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