A161881 a(n) = Sum_{k=0..n-1} {[x^k] A(x)^(n-k)} * {[x^(n-k-1)] A(x)^(k+1)} for n>0, with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.

1, 1, 2, 8, 46, 333, 2822, 26884, 280778, 3162129, 37962174, 481796692, 6424120440, 89561323131, 1300606338522, 19614272779492, 306422062160964, 4948682216714809, 82474329755007710, 1416291364674413764, 25027636488359996744, 454585893218323184316

Offset

0,3

Comments

The notation [x^n] F(x) denotes the coefficient of x^n in F(x).

Links

Table of n, a(n) for n=0..21.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 333*x^5 + 2822*x^6 +...
The table of coefficients in powers A(x)^n, n>=1, begin:
A^1: [1, 1,  2,   8,   46,   333,   2822,  26884,  280778, ...];
A^2: [1, 2,  5,  20,  112,   790,   6558,  61480,  634056, ...];
A^3: [1, 3,  9,  37,  204,  1407,  11450, 105627, 1075440, ...];
A^4: [1, 4, 14,  60,  329,  2228,  17796, 161572, 1623756, ...];
A^5: [1, 5, 20,  90,  495,  3306,  25960, 232050, 2301680, ...];
A^6: [1, 6, 27, 128,  711,  4704,  36383, 320376, 3136476, ...];
A^7: [1, 7, 35, 175,  987,  6496,  49595, 430550, 4160856, ...];
A^8: [1, 8, 44, 232, 1334,  8768,  66228, 567376, 5413977, ...];
A^9: [1, 9, 54, 300, 1764, 11619,  87030, 736596, 6942591, ...]; ...
where a(n) is obtained from the antidiagonals in the above table like so:
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;
a(7) = 1*2822 + 6*790 + 20*204 + 60*60 + 204*20 + 790*6 + 2822*1 = 26884.

Prog

(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)))}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A141117 A145844 A005840 * A219358 A088791 A111552

Adjacent sequences: A161878 A161879 A161880 * A161882 A161883 A161884

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2009

Status

approved