A151649 Number of permutations of 5 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

0, 25, 182700, 84646275, 18746073375, 3085105337250, 443146794326775, 59593466814021450, 7756190980563441400, 993121304532091347375, 126129019383244869440750, 15954027727693152179563525, 2014001281330236936094898325, 253995299147567448467485168400

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 (330,-42640,2885620,-114569650,2830998568, -44959643760,466859252380,-3196112991765,14413138051950,-42392995156008, 80058527221680,-95391910308240,69101028551520,-27758817676800,4742523426816).

Crossrefs

Column k=4 of A237202.

Programs

Magma

[(&+[(-1)^j*Binomial(5*n+1, j)*Binomial(9-j, 5)^n: j in [0..4]]): n in [1..30]]; // G. C. Greubel, Sep 12 2022

Maple

a := n -> 126^n - 56^n + 5*(5*n - 2 - 2^(n + 2)*3^n)*(25*n^3 - n)/24 + 5*7^n*n*(5*3^n*n + 3^n - 2^(3*n + 1))/2: seq(a(n), n = 1..14); # Peter Luschny, Sep 15 2022

Mathematica

Table[Sum[(-1)^j*Binomial[5*n+1, j]*Binomial[9-j, 5]^n, {j,0,4}], {n, 30}] (* G. C. Greubel, Sep 12 2022 *)

SageMath

def A151649(n): return sum((-1)^j*binomial(5*n+1, j)*binomial(9-j, 5)^n for j in (0..4))
[A151649(n) for n in (1..30)] # G. C. Greubel, Sep 12 2022

Formula

From G. C. Greubel, Sep 12 2022: (Start)

a(n) = Sum_{j=0..4} (-1)^j*binomial(5*n+1, j)*binomial(9-j, 5)^n.

G.f.: 25*x^2*(1 + 6978*x + 1016851*x^2 - 58760395*x^3 - 644809730*x^4 + 39948710783*x^5 - 333333302706*x^6 - 615347004762*x^7 + 9446377773420*x^8 - 24901972278120*x^9 + 4642437947616*x^10 + 51610957036128*x^11 + 7377258663936*x^12)/( Product_{j=0..4} (1 - binomial(j+5, 5)*x)^(5-j) ).

E.g.f.: exp(126*x) - (1 + 280*x)*exp(56*x) + (315/2)*x*(2 + 35*x)*exp(21*x) - 30*x*(4 + 75*x + 150*x^2)*exp(6*x) + (5/24)*x*(72 + 720*x + 700*x^2 + 125*x^3)*exp(x). (End)

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 06 2020