A151634 Number of permutations of 3 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

0, 0, 405, 128124, 12750255, 789300477, 38464072830, 1641724670475, 64856779908606, 2445752640197970, 89642032274378115, 3228334377697738350, 115003717118946936945, 4069184219056622926539, 143377786266629066071740, 5038841894823365860640997, 176801555321207696717476200

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 (126, -6741, 203286, -3863391, 48979386, -427502471, 2613017466, -11265590916, 34232982136, -72719412480, 106245417600, -103853184000, 64584960000, -23040000000, 3584000000).

Crossrefs

Column k=4 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, 4], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)

PARI

a(n) = {35^n - (3*n + 1)*20^n + binomial(3*n+1, 2)*10^n - binomial(3*n+1, 3)*4^n + binomial(3*n+1, 4)} \\ 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, 4) for n in (1..30)] # G. C. Greubel, Mar 26 2022

Formula

a(n) = 35^n - (3*n + 1)*20^n + binomial(3*n+1, 2)*10^n - binomial(3*n+1, 3)*4^n + binomial(3*n+1, 4). - Andrew Howroyd, May 07 2020

a(n) = Sum_{j=0..6} (-1)^j*binomial(3*n+1, 6-j)*(binomial(j+1, 3))^n. - G. C. Greubel, Mar 26 2022

Extensions

Terms a(9) and beyond from Andrew Howroyd, May 07 2020