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A003162 A binomial coefficient summation.
(Formerly M2597)
2
1, 1, 1, 3, 6, 19, 49, 163, 472, 1626, 5034, 17769, 57474, 206487, 688881, 2508195, 8563020, 31504240, 109492960, 406214878, 1432030036, 5349255726, 19077934506, 71672186953, 258095737156, 974311431094, 3537275250214, 13408623649893 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..27.

H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

FORMULA

G.f.: hypergeometric expression with an antiderivative, see Maple program. - Mark van Hoeij, May 06 2013

Recurrence: 4*n*(n+1)^2*(196*n^3 - 819*n^2 + 530*n + 528)*a(n) = 2*n*(1372*n^4 - 3633*n^3 - 7455*n^2 + 21934*n - 8448)*a(n-1) + (12740*n^6 - 90867*n^5 + 195310*n^4 - 13277*n^3 - 452690*n^2 + 528384*n - 174960)*a(n-2) + 8*(n-2)*(686*n^4 - 3010*n^3 + 1176*n^2 + 6543*n - 4725)*a(n-3) - 16*(n-3)^2*(n-2)*(196*n^3 - 231*n^2 - 520*n + 435)*a(n-4). - Vaclav Kotesovec, Mar 06 2014

a(n) ~ 4^(n+2)/(9*Pi*n^2). - Vaclav Kotesovec, Mar 06 2014

MAPLE

H := hypergeom([1/2, 1/2], [1], 16*x^2);

ogf := (Int(6*H*(4*x^2+5)/(4-x^2)^(3/2), x)+H*(16*x^2-1)/(4-x^2)^(1/2))*((2-x)/(2+x))^(1/2)/(4*x)+1/(8*x);

series(ogf, x=0, 20);  # Mark van Hoeij, May 06 2013

MATHEMATICA

Table[Sum[(Binomial[n, k]-Binomial[n, k-1])^3/Binomial[n, Floor[n/2]], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)

PROG

(PARI) a(n)=if(n<0, 0, sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^3)/binomial(n, n\2)) /* Michael Somos, Jun 02 2005 */

CROSSREFS

Cf. A003161.

Sequence in context: A148566 A148567 A148568 * A179277 A129417 A132335

Adjacent sequences:  A003159 A003160 A003161 * A003163 A003164 A003165

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 20 21:49 EDT 2018. Contains 316404 sequences. (Running on oeis4.)