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A002544
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a(n)=binomial(2*n+1,n)*(n+1)^2.
(Formerly M4855 N2075)
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6
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1, 12, 90, 560, 3150, 16632, 84084, 411840, 1969110, 9237800, 42678636, 194699232, 878850700, 3931426800, 17450721000, 76938289920, 337206098790, 1470171918600, 6379820115900, 27569305764000, 118685861314020
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Coefficients for numerical differentiation.
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REFERENCES
| C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.
J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
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).
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
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FORMULA
| G.f.: (1 + 2x)/(1 - 4x)^5/2.
a(n-1)=sum(i1+i2+...+in) where the sum is over 0<=i1<=i2<=...<=in<=n. a(n)=(n+1)^2 C(2n+1, n) G.f.: (1 + 2x)/(1 - 4x)^(5/2). - David Callan (callan(AT)stat.wisc.edu), Nov 20 2003
(n^2)*(binomial(2*n,n))/2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2006
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MAPLE
| [seq ((n^2)*(binomial(2*n, n))/2, n=1..29)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2006
a:=n->sum(sum(binomial(2*n, n)/2, j=1..n), k=1..n): seq(a(n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2007
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MATHEMATICA
| Table[Binomial[2n+1, n](n+1)^2, {n, 0, 20}] (* From Harvey P. Dale, Mar 23 2011 *)
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PROG
| (PARI) a(n)=binomial(2*n+1, n)*(n+1)^2
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CROSSREFS
| Cf. A085373.
Equals A002736/2
Sequence in context: A022640 A090749 A130592 * A093801 A135173 A173223
Adjacent sequences: A002541 A002542 A002543 * A002545 A002546 A002547
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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