A125222 a(n) = Sum_{k=0..n} C(n,k)* [x^(n-k)] A(x)^k for n>0, with a(0)=1.

1, 1, 3, 16, 119, 1116, 12522, 162863, 2404103, 39673456, 723567188, 14452217803, 313777135454, 7358812996185, 185417876158777, 4995923835850536, 143354000575456167, 4364618600823015848, 140542706037271723068

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

G.f. A(x) satisfies [from Paul D. Hanna, Jun 29 2013]:

exp( Integral (A(x) - 1)/x dx ) = (1/x)*Series_Reversion(x/(1 + x*A(x)).

Example

A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1116*x^5 + 12522*x^6 +...
From the table of n-th self-convolutions:
A^0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
A^1: [1, 1, 3, 16, 119, 1116, 12522, 162863, 2404103, 39673456, ...];
A^2: [1, 2, 7, 38, 279, 2566, 28246, 361274, 5258937, 85798608, ...];
A^3: [1, 3, 12, 67, 489, 4425, 47844, 601923, 8639097, ...];
A^4: [1, 4, 18, 104, 759, 6780, 72106, 892660, 12631271, ...];
A^5: [1, 5, 25, 150, 1100, 9731, 101955, 1242665, 17336065, ...];
A^6: [1, 6, 33, 206, 1524, 13392, 138463, 1662636, 22870059, ...];
illustrate a(n) = Sum_{k=0..n} C(n,k)*[x^(n-k)] A(x)^k by:
a(1) = 1*(0) + 1*(1) = 1;
a(2) = 1*(0) + 2*(1) + 1*(1) = 3;
a(3) = 1*(0) + 3*(3) + 3*(2) + 1*(1) = 16;
a(4) = 1*(0) + 4*(16) + 6*(7) + 4*(3) + 1*(1) = 119;
a(5) = 1*(0) + 5*(119) + 10*(38) + 10*(12) + 5*(4) + 1*(1) = 1116.
ALTERNATE FORMULA.
Define B(x) = (1/x)*Series_Reversion(x/(1 + x*A(x)),
then B(x) satisfies:
. B'(x)/B(x) = (A(x) - 1)/x;
. B(x) = 1 + x*B(x) * A(x*B(x));
. B( x/(1 + x*A(x)) ) = 1 + x*A(x).
Explicitly, B(x) begins:
B(x) = 1 + x + 2*x^2 + 7*x^3 + 37*x^4 + 265*x^5 + 2394*x^6 + 26033*x^7 +...
Note that
log(B(x)) = x + 3*x^2/2 + 16*x^3/3 + 119*x^4/4 + 1116*x^5/5 + 12522*x^6/6 +...

Prog

(PARI) {a(n)=local(A=1+sum(k=1, n-1, a(k)*x^k)); if(n==0, 1, sum(k=0, n, binomial(n, k)*polcoeff(A^k, n-k)))}

Crossrefs

Cf. A125223, A222658.

Sequence in context: A220352 A333682 A125807 * A199670 A192783 A136168

Adjacent sequences: A125219 A125220 A125221 * A125223 A125224 A125225

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2006

Status

approved