A151650 Number of permutations of 5 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.

0, 1, 273504, 554083761, 359033166276, 146006641259682, 47460662479108620, 13737399319828223622, 3735696667164317656002, 981723633332192745554763, 253168802147494901125791536, 64610303181638008483181729583, 16394452997496205694559810296928

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 (792,-260325,47435950,-5384570670,405212887824, -20987115153218,766141819691700,-20020244758559925,378098953185430160, -5187747941307534441,51805847315420838702,-376119699869936994300, 1976772497606552079000,-7462030652722893940560,20005194459069943157664, -37570195548784110639168,48620286446529148444800,-42255660508281014054400, 23493226909575103672320,-7540975791513246007296,1062524964753334075392).

Crossrefs

Column k=5 of A237202.

Programs

Magma

[(&+[(-1)^j*Binomial(5*n+1, j)*Binomial(10-j, 5)^n: j in [0..5]]): n in [1..30]]; // G. C. Greubel, Sep 12 2022

Mathematica

Table[Sum[(-1)^j*Binomial[5*n+1, j]*Binomial[10-j, 5]^n, {j,0,5}], {n, 30}] (* G. C. Greubel, Sep 12 2022 *)

SageMath

def A151650(n): return sum((-1)^j*binomial(5*n+1, j)*binomial(10-j, 5)^n for j in (0..5))
[A151650(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..5} (-1)^j*binomial(5*n+1, j)*binomial(10-j, 5)^n.

G.f.: x^2*(1 + 272712*x + 337728918*x^2 - 8648679586*x^3 - 7074308846715*x^4 + 477518433272082*x^5 - 648991861481176*x^6 - 633499131060575994*x^7 + 16047431893076948868*x^8 - 110517528944798318070*x^9 - 876288066908074857792*x^10 + 15640555372102290699216*x^11 - 63236662801921291034016*x^12 + 68632685088119444662272*x^13 + 271292956749261231644160*x^14 - 1029875415092135380492416*x^15 + 783701219351671172149248*x^16 + 907411282152901890158592*x^17 + 66939072779460046749696*x^18)/( Product_{j=0..5} (1 - binomial(j+5, 5)*x)^(6-j) ).

E.g.f.: exp(252*x) - (1 + 630*x)*exp(126*x) + 280*x*(3 + 140*x)*exp(56*x) - (105/2)*x*(8 + 525*x + 3675*x^2)*exp(21*x) + 90*x*(1 + 60*x + 350*x^2 + 375*x^3)*exp(6*x) - (1/24)*x*(144 + 5400*x + 12000*x^2 + 5625*x^3 + 625*x^4)*exp(x). (End)

Extensions

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