0,3

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.

(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)))}

Cf. A125223.

Sequence in context: A111555 A074522 A125807 this_sequence A136168 A120015 A003692

Adjacent sequences: A125219 A125220 A125221 this_sequence A125223 A125224 A125225

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 24 2006