OFFSET
0,3
COMMENTS
From Paul D. Hanna, Nov 15 2008: (Start)
More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
FORMULA
G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687.
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = g.f. of A121687. - Paul D. Hanna, Nov 08 2008
a(n) = [x^(n-1)] (2/(n+1))*A(x)^(n+1) for n>=1 with a(0)=1; i.e., a(n) equals 2/(n+1) times the coefficient of x^(n-1) in A(x)^(n+1) for n>=1. - Paul D. Hanna, Nov 15 2008
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + ...
A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...
A(x) = 1 + x*G(x)^2 where
G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + ...
is the g.f. of A121687.
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the following array.
Let A denote this sequence.
Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row:
[1, 1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, ...];
[1, 2, 5, 18, 86, 502, 3387, 25496, 209242, 1843134, 17235671, ...];
[2, 7, 27, 128, 727, 4763, 34912, 280006, 2418537, 22240055, ...];
[7, 34, 169, 958, 6173, 44364, 349152, 2965098, 26864357, ...];
[34, 203, 1195, 7707, 54792, 425216, 3560600, 31842929, ...];
[203, 1398, 9308, 66310, 510689, 4231188, 37425922, ...];
[1398, 10706, 78414, 605401, 4987185, 43742924, 406387957, ...];
[10706, 89120, 705227, 5824356, 50853813, 469182452, ...];
[89120, 794347, 6707823, 58712463, 539651646, 5211277285, ...];
[794347, 7502170, 67008980, 617340184, 5942316416, 59827126712, ...]; ...
PROG
(PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n)))/x; F=1+x*subst(F, x, x*G)^2); polcoeff(F, n)}
(PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n)))/x; F=1+x*G^2); polcoeff(F, n)} \\ Paul D. Hanna, Nov 08 2008
(PARI) /* This sequence is generated when k=1, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=1, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))} \\ Paul D. Hanna, Nov 15 2008
(PARI) /* Prints terms 0..30 */
{A=[1];
for(m=1, 30,
B=A;
for(i=1, m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1, n, C[n+1]) );
A=concat(A, 0); A[#A]=B[1]
);
A} \\ Paul D. Hanna, Jan 10 2016
(PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1)))[m+1] - Vec(Ser(A))[m+1])/(m+1)); A[n+1]}
for(n=0, 30, print1(2^n*a(n), ", ")) \\ Vaclav Kotesovec, Jan 31 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 05 2008
STATUS
approved