A001298 - OEIS

%I M5222 N2272 #78 Jun 25 2023 02:33:29

%S 0,1,31,301,1701,6951,22827,63987,159027,359502,752752,1479478,

%T 2757118,4910178,8408778,13916778,22350954,34952799,53374629,79781779,

%U 116972779,168519505,238929405,333832005,460192005,626551380,843303006,1122998436,1480692556

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

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001298/b001298.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling numbers of the 2nd kind</a>.

%H <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 G.f.: x(1 + 22x + 58x^2 + 24x^3)/(1 - x)^9. - _Paul Barry_, Aug 05 2004

%F 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 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 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 a(n) = A000915(-4-n) for all n in Z. - _Michael Somos_, Sep 04 2017

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

%t Table[StirlingS2[n+4, n], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)

%t 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 (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 (Sage) [stirling_number2(n+4,n) for n in range(0, 24)] # _Zerinvary Lajos_, May 16 2009

%o (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 Cf. A001296, A001297, A008277, A008517, A094262.

%Y Cf. A000915.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Name edited and initial zero added by _Nathaniel Johnston_, Apr 30 2011