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A113132
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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 = 5.
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6
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1, 1, 5, 50, 775, 16250, 426750, 13402500, 488566875, 20249281250, 939823431250, 48278138937500, 2719288331093750, 166652371531562500, 11040797013538437500, 786338134640203125000, 59916445436152444921875
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| a(n+1) = Sum{k, 0<=k<=n} 5^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quintic factorials (A008548).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of quintic factorials (A008548).
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EXAMPLE
| a(2) = 5.
a(3) = 2*5^2 = 50.
a(4) = 5*3*50 + 1*5*5 = 775.
a(5) = 5*4*775 + 1*5*50 + 2*50*5 = 16250.
a(6) = 5*5*16250 + 1*5*775 + 2*50*50 + 3*775*5 = 426750.
G.f.: A(x) = 1 + x + 5*x^2 + 50*x^3 + 775*x^4 + 16250*x^5 +...
= x/series_reversion(x + x^2 + 6*x^3 + 66*x^4 + 1056*x^5
+...).
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MATHEMATICA
| x=5; 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, 17}](Robert G. Wilson v (rgwv(AT)rgwv.com))
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PROG
| (PARI) {a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, if(k==1, 1, prod(j=0, k-2, 5*j+1))))))[n+1]}
(PARI) {a(n, x=5)=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
| Cf. A008548, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113133(x=6), A113134(x=7), A113135(x=8).
Sequence in context: A049393 A047054 A052562 * A088992 A116906 A051893
Adjacent sequences: A113129 A113130 A113131 * A113133 A113134 A113135
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KEYWORD
| nonn
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2005
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