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Search: id:A113135
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| A113135 |
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a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 8. |
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8
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| 1, 1, 8, 128, 3136, 103424, 4270080, 211107840, 12135936000, 794618298368, 58355305676800, 4749550536359936, 424336070117163008, 41287521140173963264, 4346005245162898325504, 492102089936714946576384
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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a(n+1) = Sum{k, 0<=k<=n} 8^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 8-fold factorials.
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 8-fold factorials.
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EXAMPLE
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a(2) = 8.
a(3) = 2*8^2 = 128.
a(4) = 8*3*128 + 1*8*8 = 3136.
a(5) = 8*4*3136 + 1*8*128 + 2*128*8 = 103424.
a(6) = 8*5*103424 + 1*8*3136 + 2*128*128 + 3*3136*8 = 4270080
G.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3136*x^4 + 103424*x^5 +...
= x/series_reversion(x + x^2 + 9*x^3 + 153*x^4 + 3825*x^5 +...).
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MATHEMATICA
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x=8; a[0]=a[1]=1; a[2]=x; a[3]=2x^2; a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}]; Table[a[n], {n, 0, 16}](Robert G. Wilson v (rgwv(AT)rgwv.com))
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PROGRAM
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(PARI) {a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, if(k==1, 1, prod(j=0, k-2, 8*j+1))))))[n+1]}
(PARI) {a(n, x=8)=if(n<0, 0, if(n==0|n==1, 1, if(n==2, x, if(n==3, 2*x^2,
x*(n-1)*a(n-1)+sum(j=2, n-2, (j-1)*a(j)*a(n-j))))))}
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CROSSREFS
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Cf. A045755, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7).
Sequence in context: A013777 A156270 A051189 this_sequence A104997 A027951 A041115
Adjacent sequences: A113132 A113133 A113134 this_sequence A113136 A113137 A113138
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KEYWORD
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nonn
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2005
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