0,3

Define A_n(x) to be a function that satisfies:

(A_n)^n = (1/x)*series_reversion( x/A(x)^n )

with A_0(x) = A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 148*x^5 +...

then A(x) = Sum_{k>=0} x^k*(A_k)^k

where a table of coefficients in (A_n(x))^n begins:

(A_0)^0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

(A_1)^1: [1, 1, 3, 13, 71, 459, 3385, 27789, 249586, ...];

(A_2)^2: [1, 2, 9, 54, 382, 3024, 26069, 240878, 2362064, ...];

(A_3)^3: [1, 3, 18, 139, 1233, 11970, 124010, 1351011, 15335562, ...];

(A_4)^4: [1, 4, 30, 284, 3049, 35460, 436172, 5594924, 74175040, ...];

(A_5)^5: [1, 5, 45, 505, 6380, 86841, 1244550, 18526345, 283982210,.];

(A_6)^6: [1, 6, 63, 818, 11901, 185940, 3051002, 51889338, ...];

(A_7)^7: [1, 7, 84, 1239, 20412, 360360, 6670888, 127835135, ...];

(A_8)^8: [1, 8, 108, 1784, 32838, 646776, 13343516, 284744600, ...];

The antidiagonal sums of the above table form this sequence.

The functions A_n(x) for n=0..10 are given by:

A_0: [1, 1, 2, 6, 26, 148, 1016, 7978, 69453, 657225, 6670401, ...];

A_1: [1, 1, 3, 13, 71, 459, 3385, 27789, 249586, 2421055, ...];

A_2: [1, 1, 4, 23, 160, 1260, 10870, 100849, 994923, 10361063, ...];

A_3: [1, 1, 5, 36, 309, 2951, 30338, 329666, 3744888, 44160033, ...];

A_4: [1, 1, 6, 52, 534, 6057, 73408, 933368, 12312793, ...];

A_5: [1, 1, 7, 71, 851, 11228, 157747, 2317332, 35202126, ...];

A_6: [1, 1, 8, 93, 1276, 19239, 308366, 5159620, 89135255, ...];

A_7: [1, 1, 9, 118, 1825, 30990, 558916, 10514222, 204056692, ...];

A_8: [1, 1, 10, 146, 2514, 47506, 952984, 19931106, 429873061, ...];

Each of these functions are related by:

A_n = A_m(x*(A_n)^(n-m)) and A_n = A(x*(A_n)^n).

(PARI) {a(n)=local(A=1+sum(k=1, n-1, a(k)*x^k)); if(n==0, 1, sum(k=0, n-1, polcoeff((serreverse(x/(A+x^2*O(x^n))^(n-k))/x), k)))}

Sequence in context: A120574 A002710 A132609 this_sequence A052844 A052859 A103937

Adjacent sequences: A125221 A125222 A125223 this_sequence A125225 A125226 A125227

nonn

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