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

1,3

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).

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

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 *)

Prog

(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

Crossrefs

Column k=4 of A174266.

Sequence in context: A185839 A285924 A285860 * A368367 A132362 A371900

Adjacent sequences: A151631 A151632 A151633 * A151635 A151636 A151637

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

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

Status

approved