0,3

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 333*x^5 +...

Coefficients in powers of A(x) begin:

A^1: [1, 1, 2, 8, 46, 333, 2822, 26884, 280778, ...];

A^2: [1, 2, 5, 20, 112, 790, 6558, 61480, ...];

A^3: [1, 3, 9, 37, 204, 1407, 11450, 105627, ...];

A^4: [1, 4, 14, 60, 329, 2228, 17796, 161572, ...];

A^5: [1, 5, 20, 90, 495, 3306, 25960, 232050, ...];

A^6: [1, 6, 27, 128, 711, 4704, 36383, 320376, ...];

A^7: [1, 7, 35, 175, 987, 6496, 49595, 430550, ...];

A^8: [1, 8, 44, 232, 1334, 8768, 66228, 567376, ...];

A^9: [1, 9, 54, 300, 1764, 11619, 87030, 736596, ...];

...

The recurrence involves products of the antidiagonals:

a(1) = 1*1 =1,

a(2) = 1*1 + 1*1 = 2,

a(3) = 1*2 + 2*2 + 2*1 = 8,

a(4) = 1*8 + 3*5 + 5*3 + 8*1 = 46,

a(5) = 1*46 + 4*20 + 9*9 + 20*4 + 46*1 = 333,

a(6) = 1*333 + 5*112 + 14*37 + 37*14 + 112*5 + 333*1 = 2822.

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

Sequence in context: A141117 A145844 A005840 this_sequence A088791 A111552 A128085

Adjacent sequences: A161878 A161879 A161880 this_sequence A161882 A161883 A161884

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 21 2009