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A147664 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^2. 8
1, 1, 3, 15, 100, 801, 7296, 73174, 791751, 9116613, 110640310, 1405349658, 18585016509, 254855278565, 3612425924919, 52793266545585, 793851646358364, 12261570084250926, 194260753173421656, 3153098224666860712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

FORMULA

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) is the g.f. of A182953.

a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 3*A(x)^(n+2)/(n+2) for n>=1 (see comment).

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 100*x^4 + 801*x^5 + 7296*x^6 +...

A(x)^2 = 1 + 2*x + 7*x^2 + 36*x^3 + 239*x^4 + 1892*x^5 + 17019*x^6 +...

A(x/A(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 239*x^5 + 1892*x^6 +...

A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) is the g.f. of A182953:

G(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 + 18178*x^6 +...

To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2),

form a table of coefficients in A(x)^(n+2) as follows:

A^3: [(1), 3, 12, 64, 426, 3345, 29766, 291999, ...];

A^4: [1, (4), 18, 100, 671, 5244, 46248, 449264, ...];

A^5: [1, 5, (25), 145, 985, 7686, 67305, 648085, ...];

A^6: [1, 6, 33, (200), 1380, 10782, 93922, 897402, ...];

A^7: [1, 7, 42, 266, (1869), 14658, 127246, 1207753, ...];

A^8: [1, 8, 52, 344, 2466, (19456), 168604, 1591496, ...];

A^9: [1, 9, 63, 435, 3186, 25335, (219522), 2063052, ...]; ...

in which the main diagonal forms the initial terms of this sequence:

[3/3*(1), 3/4*(4), 3/5*(25), 3/6*(200), 3/7*(1869), 3/8*(19456), ...].

ALTERNATE GENERATING METHOD.

This sequence forms column zero in the follow array.

Let A denote this sequence, and A^2 the self-convolution square of A.

Start in row zero with A^2, 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, 2, 7, 36, 239, 1892, 17019, 168746, 1807656, 20634852, 248560373, ...];

[1, 3, 12, 64, 426, 3345, 29766, 291999, 3097746, 35059659, 419160576,...];

[3, 15, 85, 571, 4443, 38952, 376090, 3930156, 43875903, 518191486, ...];

[15, 100, 701, 5494, 47883, 457451, 4724372, 52138050, 609215321, ...];

[100, 801, 6495, 56980, 542331, 5558082, 60793521, 704009142, ...];

[801, 7296, 65878, 630811, 6448878, 70124397, 806356465, 9749112990, ...];

[7296, 73174, 718577, 7386763, 80183478, 917721557, 11031656810, ...];

[73174, 791751, 8324862, 90823582, 1038161379, 12431200320, 155525155360, ...]; ...

PROG

(PARI) {a(n)=local(F=1+x, G); for(i=0, n, G=serreverse(x/(F+x*O(x^n))^1)/x; F=1+x*G^3); polcoeff(F, n)}

(PARI) /* This sequence is generated when k=1, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */

{a(n, k=1, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}

for(n=0, 20, print1(a(n), ", "))

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

{A=[1];

for(m=1, 30,

  B=Vec(Ser(A)^2);

  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

CROSSREFS

Cf. A182953, A145348, A145350, A120972, A145345, A145349.

Sequence in context: A255806 A226515 A135883 * A219779 A303651 A350954

Adjacent sequences:  A147661 A147662 A147663 * A147665 A147666 A147667

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 09 2008

STATUS

approved

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Last modified May 16 13:21 EDT 2022. Contains 353704 sequences. (Running on oeis4.)