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

Author

R. H. Hardin, May 29 2009

Keywords

• nonn

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).

Crossrefs

Column k=3 of A174266.

Programs

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 *)

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

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

Extensions

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