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A227995 Alternate partial sums of the binomial coefficients C(4*n,n). 2
1, 3, 25, 195, 1625, 13879, 120717, 1063323, 9454977, 84688303, 762972225, 6906366907, 62762167561, 572251392039, 5232480571761, 47961608620959, 440565328458621, 4054586252967027, 37377503512616413, 345083448151227987, 3190232694060946333, 29529002023029712547 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generally (for p>1), alternate partial sums of the binomial coefficients C(p*n,n) is 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) = (229*n^3 - 357*n^2 + 170*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)/(283*sqrt(Pi*n)*3^(3*n+1/2)).

MATHEMATICA

Table[Sum[Binomial[4*k, k]*(-1)^(n-k), {k, 0, n}], {n, 0, 20}]

PROG

(PARI) for(n=0, 50, print1(sum(k=0, n, binomial(4*k, k)), ", ")) \\ G. C. Greubel, Apr 03 2017

CROSSREFS

Cf. A054108(n-1) (p=2), A188676 (p=3), A227996 (p=5).

Sequence in context: A303602 A000544 A221777 * A037776 A037664 A289164

Adjacent sequences:  A227992 A227993 A227994 * A227996 A227997 A227998

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Aug 06 2013

STATUS

approved

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Last modified July 24 23:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)