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A037967 (C(2*n,n)^2+C(2*n,n))/2. 1
1, 3, 21, 210, 2485, 31878, 427350, 5891028, 82824885, 1181976510, 17067482146, 248817506028, 3656231188246, 54086245380300, 804670817838300, 12030722583033960, 180648817921816245, 2722858996178147310, 41179040361190612650, 624643836563467851900 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, Eq. (3.82), page 31.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..100

FORMULA

Sum_{k=0..n} (-1)^k*C(2n, k)^2 = (1/2)*(-1)^n*{ C(2n, n)^2+C(2n, n) }.

Conjecture: n^2*(n-1)*(3*n-5)*a(n) -2*(n-1)*(2*n-1)*(15*n^2-31*n+12)*a(n-1) +8*(2*n-1)*(3*n-2)*(2*n-3)^2*a(n-2)=0. - R. J. Mathar, Jul 26 2015

MATHEMATICA

Table[(Binomial[2 n, n]^2 + Binomial[2 n, n])/2, {n, 0, 45}] (* Vincenzo Librandi, Jun 02 2015 *)

PROG

(Python)

from gmpy2 import bincoef

def A037967(n):

....return bincoef(bincoef(2*n, n)+1, 2) # Chai Wah Wu, Jun 01 2015

(MAGMA) [(Binomial(2*n, n)^2+Binomial(2*n, n))/2: n in [0..30]]; // Vincenzo Librandi, Jun 02 2015

(PARI) a(n)=binomial(binomial(2*n, n)+1, 2) \\ Charles R Greathouse IV, Jun 02 2015

CROSSREFS

Sequence in context: A136223 A114469 A097690 * A123691 A087918 A088926

Adjacent sequences:  A037964 A037965 A037966 * A037968 A037969 A037970

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 22 21:45 EST 2018. Contains 299469 sequences. (Running on oeis4.)