A151633 Number of permutations of 3 indistinguishable copies of 1..n with exactly 3 adjacent element pairs in decreasing order.

0, 1, 760, 49682, 1722320, 45699447, 1063783164, 23119658500, 484099087156, 9930487583345, 201402352998560, 4059011173618086, 81520052344904040, 1634100242397204427, 32722001111322772660, 654870005050881521672, 13102000022780506515884

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (56, -1242, 14412, -96873, 394308, -984324, 1492224, -1330560, 640000, -128000).

Formula

a(n) = 20^n - (3*n + 1)*10^n + binomial(3*n+1, 2)*4^n - binomial(3*n+1, 3). - Andrew Howroyd, May 07 2020

a(n) = Sum_{j=0..5} (-1)^(j+1)*binomial(3*n+1, 5-j)*(binomial(j+1, 3))^n. - G. C. Greubel, Mar 26 2022

Mathematica

T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*Binomial[3*n+1, k-j+2]*(Binomial[j+1, 3])^n, {j, 0, k+2}];
Table[T[n, 3], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)

Prog

(PARI) a(n) = {20^n - (3*n + 1)*10^n + binomial(3*n+1, 2)*4^n - binomial(3*n+1, 3)} \\ Andrew Howroyd, May 07 2020
(Sage)
@CachedFunction
def T(n, k): return sum( (-1)^(k-j)*binomial(3*n+1, k-j+2)*(binomial(j+1, 3))^n for j in (0..k+2) )
[T(n, 3) for n in (1..30)] # G. C. Greubel, Mar 26 2022

Crossrefs

Column k=3 of A174266.

Sequence in context: A234892 A233879 A165629 * A022051 A107515 A151588

Adjacent sequences: A151630 A151631 A151632 * A151634 A151635 A151636

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(10) and beyond from Andrew Howroyd, May 07 2020

Status

approved