A151642 Number of permutations of 4 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

0, 1, 13840, 4961755, 733059110, 75073622025, 6438673851876, 503519287150295, 37463016470769170, 2712124797724710645, 193396524783642727120, 13675857973300537321251, 962624331855762939745950, 67586399804656292725004385, 4738724382451462432861849980

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 (210, -17985, 836310, -23627805, 429628026, -5189886205, 42366601950, -235447933875, 889918833750, -2267731621875, 3835990781250, -4208760859375, 2865761718750, -1098193359375, 180878906250).

Crossrefs

Column k=4 of A236463.

Programs

Magma

[(&+[(-1)^j*Binomial(4*n+1,j)*Binomial(8-j,4)^n: j in [0..4]]): n in [1..30]]; // G. C. Greubel, Sep 09 2022

Mathematica

Table[Sum[(-1)^j*Binomial[4*n+1,j]*Binomial[8-j,4]^n, {j,0,4}], {n, 30}] (* G. C. Greubel, Sep 09 2022 *)

PARI

a(n) = {70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4)} \\ Andrew Howroyd, May 07 2020

SageMath

def A151642(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n for j in (0..4))
[A151642(n) for n in (1..30)] # G. C. Greubel, Sep 09 2022

Formula

a(n) = 70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4). - Andrew Howroyd, May 07 2020

From G. C. Greubel, Sep 09 2022: (Start)

a(n) = Sum_{j=0..4} (-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n.

G.f.: x^2*(1 +13630*x +2073340*x^2 -60833350*x^3 -1182529995*x^4 [phone]*x^5 -173276304000*x^6 -651083647500*x^7 [phone]*x^8 -9980105906250*x^9 -2825648437500*x^10 [phone]*x^11 [phone]*x^12)/( Product_{j=0..4} (1 - binomial(j+4,4)*x)^(5-j) ).

E.g.f.: exp(70*x) -(1+140*x)*exp(35*x) +150*x*(1+12*x)*exp(15*x) -(50/3)*x*(3 +48*x +80*x^2)*exp(5*x) +(1/3)*x*(15 +174*x +176*x^2 +32*x^3)*exp(x). (End)

Extensions

Terms a(8) and beyond from Andrew Howroyd, May 07 2020