A113131 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 = 4.

1, 1, 4, 32, 400, 6784, 144128, 3658752, 107686656, 3599697920, 134617038848, 5567255822336, 252278661832704, 12431395516383232, 661885541595873280, 37869659304097218560, 2317293119684500193280, 151022143036329696952320

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

a(n+1) = Sum{k, 0<=k<=n} 4^k*A113129(n, k).

G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quartic factorials (A007696).

G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007696).

Example

a(2) = 4.
a(3) = 2*4^2 = 32.
a(4) = 4*3*32 + 1*4*4 = 400.
a(5) = 4*4*400 + 1*4*32 + 2*32*4 = 6784.
a(6) = 4*5*6784 + 1*4*400 + 2*32*32 + 3*400*4 = 144128.
G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 400*x^4 + 6784*x^5 +...
= x/series_reversion(x + x^2 + 5*x^3 + 45*x^4 + 585*x^5 +...).

Mathematica

x=4; 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, 4*j+1))))))[n+1]
(PARI) a(n, x=4)=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. A007696, A075834(x=1), A111088(x=2), A113130(x=3), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

Sequence in context: A005263 A325574 A373027 * A367376 A195762 A127670

Adjacent sequences: A113128 A113129 A113130 * A113132 A113133 A113134

Keyword

nonn,changed

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Status

approved