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A113130
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 = 3.
6
1, 1, 3, 18, 171, 2214, 35910, 694980, 15567795, 395396478, 11218141170, 351527039676, 12056563337598, 449255267318844, 18074052522890604, 780881956274215944, 36062953309417344579, 1772992806860541951342
OFFSET
0,3
FORMULA
a(n+1) = Sum{k, 0<=k<=n} 3^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of triple factorials (A007559).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007559).
EXAMPLE
a(2) = 3.
a(3) = 2*3^2 = 18.
a(4) = 3*3*18 + 1*3*3 = 171.
a(5) = 3*4*171 + 1*3*18 + 2*18*3 = 2214.
a(6) = 3*5*2214 + 1*3*171 + 2*18*18 + 3*171*3 = 35910.
G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 171*x^4 + 2214*x^5 +...
= x/series_reversion(x + x^2 + 4*x^3 + 28*x^4 + 280*x^5 +...).
MATHEMATICA
x=3; 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, 18}](Robert G. Wilson v)
PROG
(PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, if(k==1, 1, prod(j=0, k-2, 3*j+1))))))[n+1]
(PARI) {a(n, x=3)=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))))))}
CROSSREFS
Cf. A007559, A075834(x=1), A111088(x=2), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).
Sequence in context: A234302 A099716 A352649 * A367373 A289683 A193098
KEYWORD
nonn
AUTHOR
STATUS
approved