A335649 a(n) is the frequency of multi-pairs in a sequence of multi-set designs with 2 blocks.

0, 10, 200, 3040, 43320, 607050, 8468880, 118007680, 1643826800, 22896269770, 318906570840, 4441805503200, 61866406977960, 861688028423050, 12001766499380000, 167163044860403200, 2328280868627854560, 32428769142358413450, 451674487223023755240, 6291014052348080593120

Offset

1,2

References

A. Assaf, A. Hartman, E. Mendelsohn, Multi-set Designs-Designs having blocks with repeated elements, Congressus Numerantium, 48 (1985), 7-24.

Links

Table of n, a(n) for n=1..20.

Index entries for linear recurrences with constant coefficients, signature (19,-76,76,-19,1).

Formula

a(n) = (1/12)*((2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) - 6*(2+sqrt(3))^n - 6*(2-sqrt(3))^n + 10).

a(n) = (1/3)*(A092184(n+1)*(A092184(n+1)-1)).

From Colin Barker, Jun 16 2020: (Start)

G.f.: 10*x^2*(1 + x) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)).

a(n) = 19*a(n-1) - 76*a(n-2) + 76*a(n-3) - 19*a(n-4) + a(n-5) for n>5.

(End)

Example

For V={x,y} the design for n=2 are the blocks {xxxxxy,xyyyyy}. Pair frequencies of the multi-pairs xx, yy, and xy in these 2 blocks are all a(2)=10.
A092184(3)=6, and the above example has blocks of size 6.

Mathematica

LinearRecurrence[{19, -76, 76, -19, 1}, {0, 10, 200, 3040, 43320, 607050}, 20] (* Amiram Eldar, Jun 16 2020 *)

Prog

(PARI) concat(0, Vec(10*x^2*(1 + x) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)) + O(x^20))) \\ Colin Barker, Jun 16 2020

Crossrefs

A092184(n+1) is the block size of the n-th design in the sequence.

Sequence in context: A285021 A354410 A126431 * A202436 A320671 A237025

Adjacent sequences: A335646 A335647 A335648 * A335650 A335651 A335652

Keyword

nonn,easy

Author

John P. McSorley, Jun 15 2020

Extensions

More terms from Jinyuan Wang, Jun 15 2020

Status

approved