A151650 - OEIS

%I #10 Sep 13 2022 09:37:38

%S 0,1,273504,554083761,359033166276,146006641259682,47460662479108620,

%T 13737399319828223622,3735696667164317656002,981723633332192745554763,

%U 253168802147494901125791536,64610303181638008483181729583,16394452997496205694559810296928

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

%H Andrew Howroyd, <a href="/A151650/b151650.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (792,-260325,47435950,-5384570670,405212887824, -20987115153218,766141819691700,-20020244758559925,378098953185430160, -5187747941307534441,51805847315420838702,-376119699869936994300, 1976772497606552079000,-7462030652722893940560,20005194459069943157664, -37570195548784110639168,48620286446529148444800,-42255660508281014054400, 23493226909575103672320,-7540975791513246007296,1062524964753334075392).

%F From _G. C. Greubel_, Sep 12 2022: (Start)

%F a(n) = Sum_{j=0..5} (-1)^j*binomial(5*n+1, j)*binomial(10-j, 5)^n.

%F 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) ).

%F 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)

%t 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 *)

%o (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

%o (SageMath)

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

%o [A151650(n) for n in (1..30)] # _G. C. Greubel_, Sep 12 2022

%Y Column k=5 of A237202.

%K nonn

%O 1,3

%A _R. H. Hardin_, May 29 2009

%E Terms a(7) and beyond from _Andrew Howroyd_, May 06 2020