cp's OEIS Frontend

A001297 Stirling numbers of the second kind S(n+3, n).

%I A001297 M4974 N2136 #106 Feb 16 2025 08:32:23
%S A001297 0,1,15,90,350,1050,2646,5880,11880,22275,39325,66066,106470,165620,
%T A001297 249900,367200,527136,741285,1023435,1389850,1859550,2454606,3200450,
%U A001297 4126200,5265000,6654375,8336601,10359090,12774790,15642600,19027800
%N A001297 Stirling numbers of the second kind S(n+3, n).
%D A001297 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 A001297 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
%D A001297 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A001297 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001297 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001297 T. D. Noe, <a href="/A001297/b001297.txt">Table of n, a(n) for n = 0..1000</a>
%H A001297 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 A001297 Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq., Vol. 13 (2010), Article 10.4.4, page 5.
%H A001297 Martin 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 A001297 C. Krishnamachaki, <a href="/A001296/a001296.pdf">The operator (xD)^n</a>, J. Indian Math. Soc., Vol. 15 (1923), pp. 3-4. [Annotated scanned copy]
%H A001297 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 A001297 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 A001297 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001297 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling numbers of the 2nd kind</a>.
%H A001297 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A001297 G.f.: x*(1 + 8*x + 6*x^2)/(1 - x)^7. - _Paul Barry_, Aug 05 2004
%F A001297 E.g.f. with offset -2: exp(x)*(1*(x^3)/3! + 11*(x^4)/4! + 25*(x^5)/5! + 15*(x^6)/6!). For the coefficients [1, 11, 25, 15] see triangle A112493. E.g.f.: 1/48*x*exp(x)*(x^5+22*x^4+152*x^3+384*x^2+312*x+48)/48. Above given e.g.f. differentiated twice.
%F A001297 a(n) = (binomial(n+4, n-1) - binomial(n+3, n-2))*(binomial(n+2, n-1) - binomial(n+1, n-2)). - _Zerinvary Lajos_, May 12 2006
%F A001297 a(n) = binomial(n+1, 2)*binomial(n+3, 4). - _Vladimir Shevelev_, Dec 18 2011
%F A001297 O.g.f.: D^3(x/(1-x)) = D^4(x), where D is the operator x/(1-x)*d/dx. - _Peter Bala_, Jul 02 2012
%F A001297 a(n) = A001303(-3-n) for all n in Z. - _Michael Somos_, Sep 04 2017
%F A001297 a(n) = Sum_{k=1..n} Sum_{i=1..n} i * C(k+2,k-1). - _Wesley Ivan Hurt_, Sep 21 2017
%F A001297 From _Amiram Eldar_, Jan 10 2022: (Start)
%F A001297 Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 464/9.
%F A001297 Sum_{n>=1} (-1)^(n+1)/a(n) = 260/9 - 4*Pi^2/3 - 64*log(2)/3. (End)
%F A001297 a(n) = Sum_{0<=i<=j<=k<=n} i*j*k. - _Robert FERREOL_, May 25 2022
%e A001297 a(2) = 1*1*1 + 1*1*2 + 1*2*2 + 2*2*2 = 15
%p A001297 A001297:=-(1+8*z+6*z**2)/(z-1)**7; # _Simon Plouffe_ in his 1992 dissertation, without the initial 0
%t A001297 lst={};Do[f=StirlingS2[n+3, n];AppendTo[lst, f], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%t A001297 a[ n_] := n^2 (n + 1)^2 (n + 2) (n + 3) / 48; (* _Michael Somos_, Sep 04 2017 *)
%t A001297 Table[StirlingS2[n+3,n],{n,0,30}] (* _Harvey P. Dale_, Dec 30 2019 *)
%o A001297 (PARI) {a(n) = n^2 * (n+1)^2 * (n+2) * (n+3) / 48}; /* _Michael Somos_, Sep 04 2017 */
%o A001297 (Sage) [stirling_number2(n+3,n) for n in range(0, 34)] # _Zerinvary Lajos_, May 16 2009
%o A001297 (Magma) [n^2*(n+1)^2*(n+2)*(n+3)/48: n in [0..40]]; // _Vincenzo Librandi_, Sep 22 2017
%Y A001297 Cf. A001296, A001298, A008277, A008517, A048993, A062196, A094262.
%Y A001297 Cf. A001303.
%K A001297 nonn,easy
%O A001297 0,3
%A A001297 _N. J. A. Sloane_
%E A001297 Initial zero added by _N. J. A. Sloane_, Jan 21 2008
%E A001297 Name corrected by _Nathaniel Johnston_, Apr 30 2011