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

%I A001298 M5222 N2272 #86 Feb 21 2025 23:44:36
%S A001298 0,1,31,301,1701,6951,22827,63987,159027,359502,752752,1479478,
%T A001298 2757118,4910178,8408778,13916778,22350954,34952799,53374629,79781779,
%U A001298 116972779,168519505,238929405,333832005,460192005,626551380,843303006,1122998436,1480692556
%N A001298 Stirling numbers of the second kind S(n+4, n).
%D A001298 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.
%D A001298 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A001298 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001298 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001298 T. D. Noe, <a href="/A001298/b001298.txt">Table of n, a(n) for n = 0..1000</a>
%H A001298 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A001298 M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths2/griffiths17.html">Remodified Bessel Functions via Coincidences and Near Coincidences</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
%H A001298 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 29.
%H A001298 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001298 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling numbers of the 2nd kind</a>.
%H A001298 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A001298 G.f.: x(1 + 22x + 58x^2 + 24x^3)/(1 - x)^9. - _Paul Barry_, Aug 05 2004
%F A001298 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
%F A001298 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.
%F A001298 O.g.f. is D^4(x/(1-x)), where D is the operator x/(1-x)*d/dx. - _Peter Bala_, Jul 02 2012
%F A001298 a(n) = A000915(-4-n) for all n in Z. - _Michael Somos_, Sep 04 2017
%p A001298 A001298:=-(1+22*z+58*z**2+24*z**3)/(z-1)**9; # _Simon Plouffe_ in his 1992 dissertation, without the leading 0
%t A001298 Table[StirlingS2[n+4, n], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%t A001298 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 *)
%o A001298 (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 */
%o A001298 (Sage) [stirling_number2(n+4,n) for n in range(0, 24)] # _Zerinvary Lajos_, May 16 2009
%o A001298 (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
%Y A001298 Cf. A001296, A001297, A008277, A008517, A094262.
%Y A001298 Cf. A000915.
%K A001298 nonn,easy
%O A001298 0,3
%A A001298 _N. J. A. Sloane_
%E A001298 Name edited and initial zero added by _Nathaniel Johnston_, Apr 30 2011