A055663 Number of (3,3; n,n)-partitions of a chain of length n^2 + n.

220, 4004, 43680, 371280, 2713200, 17907120, 109830336, 637408200, 3543239700, 19028509500, 99348849600, 506679132960, 2533395664800, 12454385680800, 60338017584000, 288616850776800, 1365157704174264, 6393385628146440, 29677938224482240, 136674715507484000

Offset

9,1

Comments

a (k_1,n_1; k_2,n_2)-partition of a chain C is a chain of k_1+k_2 intervals of C, k_1 being of length n_1 and k_2 of length n_2.

Links

Vincenzo Librandi, Table of n, a(n) for n = 9..200 [a(67) and a(72) corrected by Georg Fischer, May 29 2021]

Formula

a(n) = (16/3)*(2*n-7)*(2*n-9)*(2*n-11)*(2*n-13)*(n-8)*(2*n-15)!/(n*(n-1)*(n-2)*(n-8)!^2).

D-finite with recurrence: (-9*n+n^2)*a(n) - (42-26*n+4*n^2)*a(n-1) = 0, a(9) = 220 for n >= 10. - Georg Fischer, May 29 2021

a(n) ~ n^(5/2) * 2^(2*n) / (384*sqrt(Pi)). - Vaclav Kotesovec, May 29 2021

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=9} 1/a(n) = sqrt(3)*Pi/20 - 5051/18900.

Sum_{n>=9} (-1)^(n+1)/a(n) = 65*sqrt(5)*log(phi)/6 - 15731/1350, where phi is the golden ratio (A001622). (End)

Example

a(9)=220 because in the linearly ordered set {1,...,90} we can choose in 220 ways 12 successive blocks, 3 constituted of 3 consecutive elements and 9 of 9 consecutive elements.

Maple

rec:={(-9*n+n^2)*a(n) - (42-26*n+4*n^2)*a(n-1) = 0, a(9) = 220};
f:= gfun:-rectoproc(rec, a(n), remember): map(f, [$9..26]); # Georg Fischer, May 29 2021

Prog

(Magma) [16/3*(2*n-7)*(2*n-9)*(2*n-11)*(2*n-13)*(n-8)*Factorial(2*n-15)/(n*(n-1)*(n-2)*Factorial(n-8)^2): n in [9..30]]; // Vincenzo Librandi, Jun 30 2011

Crossrefs

Cf. A055660.

Sequence in context: A215491 A258537 A027797 * A333139 A022042 A095702

Adjacent sequences: A055660 A055661 A055662 * A055664 A055665 A055666

Keyword

nonn

Author

Paolo Dominici (pl.dm(AT)libero.it), Jun 07 2000

Status

approved