A001297 Stirling numbers of the second kind S(n+3, n). (Formerly M4974 N2136)

0, 1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470, 165620, 249900, 367200, 527136, 741285, 1023435, 1389850, 1859550, 2454606, 3200450, 4126200, 5265000, 6654375, 8336601, 10359090, 12774790, 15642600, 19027800

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.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.

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].

Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4, page 5.

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

C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., Vol. 15 (1923), pp. 3-4. [Annotated scanned copy]

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 (7,-21,35,-35,21,-7,1).

Formula

G.f.: x*(1 + 8*x + 6*x^2)/(1 - x)^7. - Paul Barry, Aug 05 2004

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.

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

a(n) = binomial(n+1, 2)*binomial(n+3, 4). - Vladimir Shevelev, Dec 18 2011

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

a(n) = A001303(-3-n) for all n in Z. - Michael Somos, Sep 04 2017

a(n) = Sum_{k=1..n} Sum_{i=1..n} i * C(k+2,k-1). - Wesley Ivan Hurt, Sep 21 2017

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 464/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 260/9 - 4*Pi^2/3 - 64*log(2)/3. (End)

a(n) = Sum_{0<=i<=j<=k<=n} i*j*k. - Robert FERREOL, May 25 2022

Example

a(2) = 1*1*1 + 1*1*2 + 1*2*2 + 2*2*2 = 15

Maple

A001297:=-(1+8*z+6*z**2)/(z-1)**7; # Simon Plouffe in his 1992 dissertation, without the initial 0

Mathematica

lst={}; Do[f=StirlingS2[n+3, n]; AppendTo[lst, f], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
a[ n_] := n^2 (n + 1)^2 (n + 2) (n + 3) / 48; (* Michael Somos, Sep 04 2017 *)
Table[StirlingS2[n+3, n], {n, 0, 30}] (* Harvey P. Dale, Dec 30 2019 *)

Prog

(PARI) {a(n) = n^2 * (n+1)^2 * (n+2) * (n+3) / 48}; /* Michael Somos, Sep 04 2017 */
(Sage) [stirling_number2(n+3, n) for n in range(0, 34)] # Zerinvary Lajos, May 16 2009
(Magma) [n^2*(n+1)^2*(n+2)*(n+3)/48: n in [0..40]]; // Vincenzo Librandi, Sep 22 2017

Crossrefs

Cf. A001296, A001298, A008277, A008517, A048993, A062196, A094262.

Cf. A001303.

Sequence in context: A151974 A179096 A328994 * A005716 A292055 A263628

Adjacent sequences: A001294 A001295 A001296 * A001298 A001299 A001300

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Initial zero added by N. J. A. Sloane, Jan 21 2008

Name corrected by Nathaniel Johnston, Apr 30 2011

Status

approved