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A225612 Partial sums of the binomial coefficients C(4*n,n). 2
1, 5, 33, 253, 2073, 17577, 152173, 1336213, 11854513, 105997793, 953658321, 8622997453, 78291531921, 713305091521, 6518037055321, 59712126248041, 548239063327621, 5043390644753269, 46475480410336709, 428936432074181109, 3964252574286355429 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).

a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).

MAPLE

A225612:=n->add(binomial(4*k, k), k=0..n): seq(A225612(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

MATHEMATICA

Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]

Accumulate[Table[Binomial[4n, n], {n, 0, 20}]] (* Harvey P. Dale, Feb 01 2015 *)

PROG

(PARI) for(n=0, 50, print1(sum(k=0, n, binomial(4*k, k)), ", ")) \\ G. C. Greubel, Apr 01 2017

CROSSREFS

Cf. A006134 (p=2), A188675 (p=3), A225615 (p=5).

Sequence in context: A056159 A171804 A324312 * A330802 A199552 A061253

Adjacent sequences:  A225609 A225610 A225611 * A225613 A225614 A225615

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Aug 06 2013

STATUS

approved

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Last modified July 8 07:34 EDT 2020. Contains 335513 sequences. (Running on oeis4.)