|
| |
|
|
A001298
|
|
Stirling numbers of the second kind S(n+4,n).
(Formerly M5222 N2272)
|
|
8
| |
|
|
0, 1, 31, 301, 1701, 6951, 22827, 63987, 159027, 359502, 752752, 1479478, 2757118, 4910178, 8408778, 13916778, 22350954, 34952799, 53374629, 79781779, 116972779, 168519505, 238929405, 333832005, 460192005, 626551380, 843303006, 1122998436, 1480692556
(list; graph; refs; listen; history; internal format)
|
|
|
|
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.
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].
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
|
|
|
FORMULA
| G.f. : x(1+22x+58x^2+24x^3)/(1-x)^9 - Paul Barry (pbarry(AT)wit.ie), Aug 05 2004
a(n) = Stirling2(n+4, n) = Sum(Sum(Sum(Sum(i*j*k*l, i = 1 .. j), j = 1 .. k), k = 1 .. l), l = 1 .. n) = (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 (vladeta(AT)eunet.rs), Jan 31 2005
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.
|
|
|
MAPLE
| A001298:=-(1+22*z+58*z**2+24*z**3)/(z-1)**9; [S. Plouffe in his 1992 dissertation, without the leading 0.]
|
|
|
MATHEMATICA
| Table[StirlingS2[n+4, n], {n, 0, 100}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 27 2008]
|
|
|
PROG
| (Sage) [stirling_number2(n+4, n) for n in xrange(0, 24)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
|
|
|
CROSSREFS
| Cf. A008277, A094262, A001296, A001297.
Sequence in context: A161558 A156094 A115151 * A027841 A000500 A141912
Adjacent sequences: A001295 A001296 A001297 * A001299 A001300 A001301
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Name edited and initial zero added by Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), Apr 30 2011
|
| |
|
|
