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A273631 a(n) = Sum_{k = 0..n} (-1)^k*binomial(k,2)^3*binomial(n,k)^3. 1
0, 0, 1, 0, -1296, 0, 303750, 0, -36879360, 0, 3157481250, 0, -217564322976, 0, 12926105848656, 0, -689598074880000, 0, 33901459248661290, 0, -1562983866658500000, 0, 68423756889802253940, 0, -2870422192164339671040, 0, 116191495035298068750000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Let d(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)^3. Clearly, by symmetry of the binomial coefficients we have d(2*n + 1) = 0. Dixon's identity is the result d(2*n) = (-1)^n*(3*n)!/n!^3. A generalization is: for r a nonnegative integer there holds Sum_{k = 0..n} (-1)^k*binomial(k,r)^3*binomial(n,k)^3 = (-1)^r*binomial(n,r)^3*d(n - r). This is the case r = 2. See A273630 (case r = 1) and A245086 (case r = 0).

LINKS

Table of n, a(n) for n=0..26.

P. Bala, A generalization of Dixon's identity

J. Ward, 100 Years of Dixon's Identity, Irish Mathematical Society Bulletin 27, 46-54, 1991

Wikipedia, Dixon's identity

FORMULA

a(0) = 0 and a(2*n + 2) = (-1)^n*binomial(2*n + 2,2)^3*(3*n)!/n!^3 for n >= 0. a(2*n + 1) = 0.

a(2*n + 2) = (-1)^n*(n + 1)^3*(2*n + 1)^3 * A006480(n) for n >= 0.

a(n) = Sum_{k = 2..n} (-1)^k*multinomial(n, 2, k - 2, n - k)^3.

Recurrence: a(n) = -3*n^3*(n - 1)^3*(3*n - 8)*(3*n - 10)/((n - 2)^5*(n - 3)^3) * a(n-2).

MAPLE

seq(add((-1)^k*binomial(k, 2)^3*binomial(n, k)^3, k = 0..n), n = 0..30);

MATHEMATICA

Table[Sum[(-1)^k*Binomial[k, 2]^3 Binomial[n, k]^3, {k, 0, n}], {n, 0, 27}] (* Michael De Vlieger, Jul 22 2016 *)

PROG

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(k, 2)^3*binomial(n, k)^3) \\ Felix Fröhlich, Jul 22 2016

(MAGMA) [&+[(-1)^k*Binomial(k, 2)^3*Binomial(n, k)^3: k in [0..n]]: n in [0..70]]; // Vincenzo Librandi, Jul 23 2016

CROSSREFS

Cf. A006480, A245086, A273630.

Sequence in context: A139027 A043392 A179536 * A186486 A186485 A281399

Adjacent sequences:  A273628 A273629 A273630 * A273632 A273633 A273634

KEYWORD

sign,easy

AUTHOR

Peter Bala, Jul 17 2016

STATUS

approved

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Last modified April 26 08:14 EDT 2019. Contains 322472 sequences. (Running on oeis4.)