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A001297
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Stirling numbers of the second kind S(n+3,n).
(Formerly M4974 N2136)
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9
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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
(list;
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history;
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internal format)
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OFFSET
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0,3
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REFERENCES
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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.
L. 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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..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].
M. 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., 15 (1923),3-4. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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).
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FORMULA
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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. is D^3(x/(1-x)) = D^4(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
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MAPLE
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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
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lst={}; Do[f=StirlingS2[n+3, n]; AppendTo[lst, f], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
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PROG
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(Sage) [stirling_number2(n+3, n) for n in xrange(0, 34)] # Zerinvary Lajos, May 16 2009
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CROSSREFS
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Cf. A001296, A001298, A008277, A008517, A048993, A062196, A094262.
Sequence in context: A022707 A151974 A179096 * A005716 A218408 A048630
Adjacent sequences: A001294 A001295 A001296 * A001298 A001299 A001300
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Gerald McGarvey, Jul 18 2004
Initial zero added by N. J. A. Sloane, Jan 21 2008
Name corrected by Nathaniel Johnston, Apr 30 2011
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STATUS
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approved
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