A075820 -id:A075820

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A243195

G.f.: A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - 4*x*F(2,x)^2)^(1/2), F(2,x) = 1/(1 - 9*x/F(3,x)^3)^(1/3), F(3,x) = 1/(1 - 16*x/F(4,x)^4)^(1/4), ..., so that F(n,x) = 1/(1 - (n+1)^2*x*F(n+1,x)^(n+1))^(1/(n+1)) for n>=0.

+10
3

1, 1, 3, 23, 373, 10625, 468449, 29464867, 2501594061, 275449029949, 38159974444647, 6494568101890209, 1331919152059028501, 323931747385690406859, 92181008940950546725329, 30343276257278261997917167, 11439766584058218615650332597, 4897281474298488766530388729025

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare to an o.g.f. E(x) for the Euler numbers (A000364):

E(x) = G(0,x) where G(0,x) = 1/(1 - x*G(1,x)^2), G(1,x) = 1/(1 - 4*x*G(2,x)^3)^(1/2), G(2,x) = 1/(1 - 9*x/G(3,x)^4)^(1/3), G(3,x) = 1/(1 - 16*x/G(4,x)^5)^(1/4), ..., so that G(n,x) = 1/(1 - (n+1)^2*x*G(n+1,x)^(n+2))^(1/(n+1)) for n>=0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..150

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1901*x^5 + 31841*x^6 +...

Define F(n,x) with F(0,x) = A(x) where for n>0 we have

F(n-1,x) = 1/(1 - n^2*x*F(n,x)^n)^(1/n), or equivalently

F(n,x) = ( (1 - 1/F(n-1,x)^n) / (n^2*x) )^(1/n),

then

F(1,x) = 1 + 2*x + 18*x^2 + 326*x^3 + 9826*x^4 + 445860*x^5 +...

F(2,x) = 1 + 3*x + 54*x^2 + 1782*x^3 + 88113*x^4 + 6006267*x^5 +...

F(3,x) = 1 + 4*x + 120*x^2 + 6280*x^3 + 465040*x^4 + 45026364*x^5 +...

F(4,x) = 1 + 5*x + 225*x^2 + 17125*x^3 + 1779800*x^4 + 233701800*x^5 +...

F(5,x) = 1 + 6*x + 378*x^2 + 39438*x^3 + 5484738*x^4 + 941166954*x^5 +...

F(6,x) = 1 + 7*x + 588*x^2 + 80556*x^3 + 14457646*x^4 + 3147092914*x^5 +...

F(7,x) = 1 + 8*x + 864*x^2 + 150432*x^3 + 33866688*x^4 + 9128319696*x^5 +...

...

F(1,x)^2 = 1 + 4*x + 40*x^2 + 724*x^3 + 21280*x^4 + 942760*x^5 +...

F(2,x)^3 = 1 + 9*x + 189*x^2 + 6345*x^3 + 306621*x^4 + 20256561*x^5 +...

F(3,x)^4 = 1 + 16*x + 576*x^2 + 31136*x^3 + 2271296*x^4 + 213398256*x^5 +...

F(4,x)^5 = 1 + 25*x + 1375*x^2 + 109375*x^3 + 11289625*x^4 + 1444554625*x^5 +...

F(5,x)^6 = 1 + 36*x + 2808*x^2 + 308988*x^3 + 42986448*x^4 + 7223051340*x^5 +...

F(6,x)^7 = 1 + 49*x + 5145*x^2 + 748769*x^3 + 135256905*x^4 + 28966780465*x^5 +...

...

PROG

(PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - (n-k)^2*x*A^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A075820, A243161.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 31 2014

STATUS

approved

A243161

G.f. A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - (2*x*F(2,x))^2)^(1/2), F(2,x) = 1/(1 - (3*x*F(3,x))^3)^(1/3), ..., so that F(n-1,x)^n = 1/(1 - (n*x*F(n,x))^n) for n>=0.

+10
2

1, 1, 1, 3, 5, 13, 61, 133, 449, 1825, 11497, 29905, 121529, 613121, 3192553, 26963653, 76748369, 367110269, 2101537105, 13742608029, 90490605353, 966603833657, 2899027074937, 15202727310033, 102739122225929, 738145449190921, 6064931145859705, 47996143247509851, 637518525737986877

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..250

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 13*x^5 + 61*x^6 + 133*x^7 + 449*x^8 + 1825*x^9 + 11497*x^10 + 29905*x^11 + 121529*x^12 +...

Define F(n,x) with F(0,x) = A(x), where for n>0 we have

F(n-1,x) = 1/(1 - (n*x*F(n,x))^n )^(1/n), or equivalently

F(n,x) = (1 - 1/F(n-1,x)^n)^(1/n) / (n*x),

then

F(1,x) = 1 + 2*x^2 + 3*2*x^4 + 6^2*x^5 + 5*2^2*x^6 + 3^3*2^3*x^7 +...

F(2,x) = 1 + 3^2*x^3 + 6*3^3*x^6 + 12^3*x^7 + 14*3^5*x^9 + 4^4*3^5*x^10 +...

F(3,x) = 1 + 4^3*x^4 + 10*4^5*x^8 + 20^4*x^9 + 30*4^8*x^12 + 5^5*4^7*x^13 +...

F(4,x) = 1 + 5^4*x^5 + 15*5^7*x^10 + 30^5*x^11 + 55*5^11*x^15 + 6^6*5^9*x^16 +...

F(5,x) = 1 + 6^5*x^6 + 21*6^9*x^12 + 42^6*x^13 + 91*6^14*x^18 + 7^7*6^11*x^19 +...

....

F(1,x)^2 = 1 + 2^2*x^2 + 2^4*x^4 + 2^3*3^2*x^5 + 2^6*x^6 + 2*3^2*2^5*x^7 +...

F(2,x)^3 = 1 + 3^3*x^3 + 3^6*x^6 + 3^4*4^3*x^7 + 3^9*x^9 + 2*4^3*3^7*x^10 +...

F(3,x)^4 = 1 + 4^4*x^4 + 4^8*x^8 + 4^5*5^4*x^9 + 4^12*x^12 + 2*5^4*4^9*x^13 +...

F(4,x)^5 = 1 + 5^5*x^5 + 5^10*x^10 + 5^6*6^5*x^11 + 5^15*x^15 + 2*6^5*5^11*x^16 +...

F(5,x)^6 = 1 + 6^6*x^6 + 6^12*x^12 + 6^7*7^6*x^13 + 6^18*x^18 + 2*7^6*6^13*x^19 +...

...

PROG

(PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - ((n-k)*x*A)^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}