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A145350 G.f. satisfies: A(x/A(x)^2) = 1 + x*A(x). 8
1, 1, 3, 18, 154, 1632, 20007, 273164, 4058556, 64628487, 1091488334, 19403175105, 361028420037, 7000932594042, 141010975529568, 2942134448306481, 63449975020918843, 1411787024678728344, 32360032648643379471, 763096191377494726161, 18491954778730596443088 (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..374

FORMULA

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).

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

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 154*x^4 + 1632*x^5 +...

A(x)^2 = 1 + 2*x + 7*x^2 + 42*x^3 + 353*x^4 + 3680*x^5 + 44526*x^6+...

A(x/A(x)^2) = 1 + x + x^2 + 3*x^3 + 18*x^4 + 154*x^5 + 1632*x^6 +...

A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)^2):

G(x) = 1 + x + 5*x^2 + 41*x^3 + 432*x^4 + 5329*x^5 + 73512*x^6 +...

G(x)^2 = 1 + 2*x + 11*x^2 + 92*x^3 + 971*x^4 + 11932*x^5 +...

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

form a table of coefficients in A(x)^(2*n+1) as follows:

A^3: [(1), 3, 12, 73, 606, 6225, 74370, 994668, ...];

A^5: [1, (5), 25, 160, 1315, 13191, 153930, 2017620, ...];

A^7: [1, 7, (42), 287, 2373, 23436, 267988, 3445835, ...];

A^9: [1, 9, 63, (462), 3888, 38106, 428637, 5414760, ...];

A^11: [1, 11, 88, 693, (5984), 58619, 651354, 8099410, ...];

A^13: [1, 13, 117, 988, 8801, (86697), 955656, 11723712, ...];

A^15: [1, 15, 150, 1355, 12495, 124398, (1365820), 16571385, ...]; ...

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

[3/3*(1), 3/5*(5), 3/7*(42), 3/9*(462), 3/11*(5984), 3/13*(86697), ...].

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 this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A^2 and the remaining terms in a given row to obtain the next row:

[1, 1, 1, 3, 18, 154, 1632, 20007, 273164, 4058556, 64628487, ...];

[1, 3, 12, 73, 606, 6225, 74370, 994668, 14535285, 228349287, ...];

[3, 18, 118, 962, 9511, 109404, 1415942, 20128565, 309001962, ...];

[18, 154, 1324, 13017, 146470, 1849625, 25701033, 386747469, ...];

[154, 1632, 16743, 188240, 2343654, 32006379, 473572975, ...];

[1632, 20007, 233150, 2905879, 39290669, 573813430, 8978918475, ...];

[20007, 273164, 3512228, 47574771, 689590692, 10679554646, ...];

[273164, 4058556, 56511375, 820798718, 12635699895, ...];

[4058556, 64628487, 962231360, 14843336308, 241004566025, ...]; ...

PROG

(PARI) {a(n)=local(A=1+x, G); for(i=0, n, G=(serreverse(x/(A+x*O(x^n))^2)/x)^(1/2); A=1+x*G^3); polcoeff(A, n)}

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

{a(n, k=2, m=1)=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=A;

  for(i=1, m-1, C=Vec(Ser(A)^2*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. variants: A001764, A147664, A120972, A145345, A145349, A147664.

Sequence in context: A200320 A207569 A005412 * A107888 A293491 A138274

Adjacent sequences:  A145347 A145348 A145349 * A145351 A145352 A145353

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 12 2008

STATUS

approved

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Last modified January 18 04:57 EST 2021. Contains 340250 sequences. (Running on oeis4.)