A243426 G.f. satisfies: a(n) = [x^n] ( A(x)^(n+1) + 1/A(x)^(n+1) ) / (n+1) for n>1, with a(0)=a(1)=1.

1, 1, 3, 20, 215, 3019, 50750, 978792, 21118641, 501447228, 12954812178, 361153740034, 10796374169997, 344373435613191, 11672904804892441, 419017404782400368, 15881588444144243375, 633896023527831462147, 26581090260348132921162, 1168473922210084561137995

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

G.f. satisfies: A(x) = x-1 + Series_Reversion(x*A(x))/x + Series_Reversion(x/A(x))/x.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 215*x^4 + 3019*x^5 + 50750*x^6 +...
Form a table of coefficients in A(x)^n + 1/A(x)^n for n>=1, which begins:
[2, 0,  1,   5,   41,   474,   6816,   115062,   2209419, ...];
[2, 0,  4,  20,  165,  1906,  27371,   461606,   8857729, ...];
[2, 0,  9,  45,  375,  4326,  61987,  1043721,  20005287, ...];
[2, 0, 16,  80,  676,  7784, 111204,  1868272,  35753349, ...];
[2, 0, 25, 125, 1075, 12350, 175785,  2944975,  56245065, ...];
[2, 0, 36, 180, 1581, 18114, 256723,  4286502,  81666879, ...];
[2, 0, 49, 245, 2205, 25186, 355250,  5908616, 112250341, ...];
[2, 0, 64, 320, 2960, 33696, 472848,  7830336, 148274340, ...];
[2, 0, 81, 405, 3861, 43794, 611262, 10074132, 190067769, ...]; ...
then this sequence is obtained from the main diagonal after the initial 2 terms:
[2, 0, 9/3, 80/4, 1075/5, 18114/6, 355250/7, 7830336/8, 190067769/9, ...].
RELATED SERIES.
Define F(x) = A(x/F(x)):
F(x) = 1 + x + 2*x^2 + 13*x^3 + 142*x^4 + 1988*x^5 + 33250*x^6 +...
1/F(x) = 1 - x - x^2 - 10*x^3 - 117*x^4 - 1696*x^5 - 29060*x^6 -...
and G(x) = A(x*G(x)):
G(x) = 1 + x + 4*x^2 + 30*x^3 + 332*x^4 + 4715*x^5 + 79810*x^6 +...
then A(x) = x-1 + G(x) + 1/F(x).

Prog

(PARI) {a(n)=local(A=vector(n+1, i, 1)); for(n=2, #A-1, A[n+1] = Vec(Ser(A)^(n+1) + 1/Ser(A)^(n+1))[n+1]/(n+1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Build vector of n=0..30 terms (faster): */
{A=vector(31, i, 1); for(n=2, #A-1, A[n+1] = Vec(Ser(A)^(n+1) + 1/Ser(A)^(n+1))[n+1]/(n+1)); A}

Crossrefs

Cf. A237644.

Sequence in context: A230478 A014068 A006963 * A365438 A113333 A206405

Adjacent sequences: A243423 A243424 A243425 * A243427 A243428 A243429

Keyword

nonn

Author

Paul D. Hanna, Jun 04 2014

Status

approved