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A242054
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Column 3 of square array A246072 / n!.
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2
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1, 1, 5, 18, 75, 396, 2052, 11586, 71787, 458352, 3103668, 22202874, 164999826, 1281692088, 10371684312, 86973240204, 755908929603, 6794220017664, 63008287321788, 602270212069098, 5924679849081126, 59897824980579576, 621672797654084520, 6616610400436719588
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OFFSET
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0,3
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LINKS
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FORMULA
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Recurrence: n*(125*n^10 - 8575*n^9 + 249165*n^8 - 3972421*n^7 + 38651424*n^6 - 241441049*n^5 + 985299581*n^4 - 2598873155*n^3 + 4233949973*n^2 - 3823560792*n + 1433318628)*a(n) = (n-1)*(250*n^10 - 17275*n^9 + 505255*n^8 - 7939307*n^7 + 74253125*n^6 - 433278950*n^5 + 1592429495*n^4 - 3581735158*n^3 + 4503495303*n^2 - 2437269642*n + 30743664)*a(n-1) + (250*n^12 - 17025*n^11 + 485380*n^10 - 7526552*n^9 + 70321082*n^8 - 411180662*n^7 + 1481140960*n^6 - 2907087063*n^5 + 1110382294*n^4 + 8175223780*n^3 - 18648517428*n^2 + 16445473512*n - 5256191136)*a(n-2) + (125*n^13 - 7575*n^12 + 179440*n^11 - 1844901*n^10 + 1377531*n^9 + 175414789*n^8 - 2147090919*n^7 + 13587878007*n^6 - 53501005344*n^5 + 136256919050*n^4 - 221250046545*n^3 + 215137237254*n^2 - 107999226624*n + 18548116944)*a(n-3) - (3125*n^12 - 230275*n^11 + 7312925*n^10 - 129375074*n^9 + 1424467574*n^8 - 10349546810*n^7 + 51109969432*n^6 - 173096865441*n^5 + 398028650726*n^4 - 601201146808*n^3 + 556233509382*n^2 - 271925267076*n + 47484061632)*a(n-4) - (n-4)*(3125*n^12 - 222875*n^11 + 6789125*n^10 - 115816245*n^9 + 1238520315*n^8 - 8794888450*n^7 + 42652555528*n^6 - 142304505773*n^5 + 322756817266*n^4 - 480186892231*n^3 + 434864239701*n^2 - 204163716870*n + 31770252360)*a(n-5) - (n-5)*(n-4)*(1375*n^12 - 95825*n^11 + 2889440*n^10 - 49131951*n^9 + 525311115*n^8 - 3732302998*n^7 + 18093070721*n^6 - 60182645063*n^5 + 135414119977*n^4 - 198015832273*n^3 + 172828363290*n^2 - 74116518432*n + 7983006192)*a(n-6) - (n-6)*(n-5)*(n-4)*(250*n^10 - 13325*n^9 + 316180*n^8 - 4021155*n^7 + 29819852*n^6 - 134332700*n^5 + 368917385*n^4 - 590456906*n^3 + 472005585*n^2 - 82277310*n - 69925968)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(250*n^10 - 14025*n^9 + 329680*n^8 - 4094477*n^7 + 29774855*n^6 - 132986602*n^5 + 367961264*n^4 - 609685740*n^3 + 541560093*n^2 - 175805898*n - 27276984)*a(n-8) - (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(125*n^10 - 7325*n^9 + 177615*n^8 - 2272801*n^7 + 17127047*n^6 - 80049056*n^5 + 235218311*n^4 - 424843636*n^3 + 437741568*n^2 - 212268240*n + 23612904)*a(n-9).
a(n) ~ n^(2*n/3) * exp(-2*n/3 + n^(2/3) + 4/3*n^(1/3) - 8/9) / sqrt(3). - Vaclav Kotesovec, Aug 13 2014
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MAPLE
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with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
proc(k, m, i, t) option remember; local d, j; d:= l[i];
`if`(i=1, m!, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
(d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
`if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
`if`(t=0, [][], m/t))))
end; g(k, n-k, nops(l), 0)
end:
A:= (n, k)-> `if`(k=0, (2*n)!, b(2*n, n, k)):
seq(A(n, 3)/n!, n = 0..20); # after Alois P. Heinz
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!);
M = multinomial;
b[n_, k_, p_] := b[n, k, p] = Module[{l, g}, l = Sort[Divisors[p]]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[{d}, d = l[[i]]; If[i == 1, m!, Sum[M[k0, Join[{k0-(d-t)j}, Table[d-t, {j}]]]/j! (d-1)!^j M[m, Join[{m - t j}, Table[t, {j}]]] If[d-t == 1, g[k0 - (d-t) j, m - t j, i-1, 0], g[k0 - (d-t)j, m - t j, i, t+1]], {j, 0, Min[k0/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k, n-k, Length[l], 0]];
A[n_, k_] := If[k == 0, (2n)!, b[2n, n, k]];
a[n_] := A[n, 3]/n!;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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