A145345 - OEIS

%I #27 Jan 31 2023 11:03:50

%S 1,1,2,7,34,203,1398,10706,89120,794347,7502170,74511150,773864654,

%T 8368430208,93905460014,1090519614152,13077315637592,161643281777801,

%U 2056306418177832,26887064722265250,360939404438509866

%N G.f. satisfies: A(x/A(x)) = 1 + x*A(x).

%C From _Paul D. Hanna_, Nov 15 2008: (Start)

%C More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then

%C A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);

%C thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)

%H Vaclav Kotesovec, <a href="/A145345/b145345.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687.

%F 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

%F 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

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + ...

%e A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...

%e A(x) = 1 + x*G(x)^2 where

%e G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + ...

%e is the g.f. of A121687.

%e ALTERNATE GENERATING METHOD.

%e This sequence forms column zero in the following array.

%e Let A denote this sequence.

%e 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:

%e [1, 1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, ...];

%e [1, 2, 5, 18, 86, 502, 3387, 25496, 209242, 1843134, 17235671, ...];

%e [2, 7, 27, 128, 727, 4763, 34912, 280006, 2418537, 22240055, ...];

%e [7, 34, 169, 958, 6173, 44364, 349152, 2965098, 26864357, ...];

%e [34, 203, 1195, 7707, 54792, 425216, 3560600, 31842929, ...];

%e [203, 1398, 9308, 66310, 510689, 4231188, 37425922, ...];

%e [1398, 10706, 78414, 605401, 4987185, 43742924, 406387957, ...];

%e [10706, 89120, 705227, 5824356, 50853813, 469182452, ...];

%e [89120, 794347, 6707823, 58712463, 539651646, 5211277285, ...];

%e [794347, 7502170, 67008980, 617340184, 5942316416, 59827126712, ...]; ...

%o (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)}

%o (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

%o (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

%o (PARI) /* Prints terms 0..30 */

%o {A=[1];

%o for(m=1,30,

%o   B=A;

%o   for(i=1,m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1,n,C[n+1]) );

%o   A=concat(A,0);A[#A]=B[1]

%o );

%o A} \\ _Paul D. Hanna_, Jan 10 2016

%o (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]}

%o for(n=0, 30, print1(2^n*a(n), ", ")) \\ _Vaclav Kotesovec_, Jan 31 2023

%Y Cf. A121687, A145350, A145349, A147664, A302703.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 05 2008