A001297 Stirling numbers of the second kind S(n+3, n).
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
Examples
a(2) = 1*1*1 + 1*1*2 + 1*2*2 + 2*2*2 = 15
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]
- 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 (7,-21,35,-35,21,-7,1).
Programs
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Magma
[n^2*(n+1)^2*(n+2)*(n+3)/48: n in [0..40]]; // Vincenzo Librandi, Sep 22 2017
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Maple
A001297:=-(1+8*z+6*z**2)/(z-1)**7; # Simon Plouffe in his 1992 dissertation, without the initial 0
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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 *) -
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
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
Extensions
Initial zero added by N. J. A. Sloane, Jan 21 2008
Name corrected by Nathaniel Johnston, Apr 30 2011