A122888 -id:A122888 - cp's OEIS frontend

A194971 a(n) equals the coefficient of x^(2*n-1) in the n-th iteration of x+x^2 for n>=1.

1, 2, 10, 188, 8994, 832680, 127104492, 28951041456, 9201410927608, 3889680139527920, 2109876998624179100, 1428197506614652750656, 1179911974067256647171268, 1168294604146384807206421176, 1365624160842343461171218423880

Offset: 1

Paul D. Hanna, Sep 06 2011

nonn

			The coefficients of x^k, k>=1, in the n-th iterations of x+x^2 begin:
n=1: [(1), 1];
n=2: [1, 2, (2), 1];
n=3: [1, 3, 6, 9, (10), 8, 4, 1];
n=4: [1, 4, 12, 30, 64, 118, (188), 258, 302, 298, 244,162,84,32,8,1];
n=5: [1, 5, 20, 70, 220, 630, 1656, 4014, (8994), 18654, 35832,...];
n=6: [1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, (832680), ...]; ...
coefficients in parenthesis form the initial terms of this sequence.

Cf. A194972, A122888, A112317.

{a(n)=local(A=x,G=x+x^2); for(i=1,n, A=subst(G, x, A+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}

A194972 a(n) equals the coefficient of x^n in the (2*n-1)-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 3, 20, 231, 3864, 85140, 2332616, 76485227, 2921536088, 127421864328, 6248486040840, 340321635330534, 20383240346962440, 1331538898625750100, 94216429100347571448, 7178425650032302557691, 585936966156456139931584, 51011156414845408925712816

Offset: 1

Paul D. Hanna, Sep 06 2011

nonn

			The coefficients of x^k, k>=1, in the odd iterations of x+x^2 begin:
n=1: [(1), 1];
n=3: [1,(3), 6, 9, 10, 8, 4, 1];
n=5: [1, 5,(20), 70, 220, 630, 1656, 4014, 8994, 18654, ...];
n=7: [1, 7, 42,(231), 1190, 5810, 27076, 121023, 520626, ...];
n=9: [1, 9, 72, 540,(3864), 26628, 177744, 1153740, 7303164, ...];
n=11:[1, 11, 110, 1045, 9570,(85140), 739332, 6286797, ...];
n=13:[1, 13, 156, 1794, 20020, 218218,(2332616), 24519066, ...];
n=15:[1, 15, 210, 2835, 37310, 481390, 6110468,(76485227), ...]; ...
coefficients in parenthesis form the initial terms of this sequence.

Cf. A194971, A122888, A112317.

{a(n)=local(A=x,G=x+x^2); for(i=1,2*n-1, A=subst(G, x, A+x*O(x^n))); polcoeff(A,n)}

A182968 G.f.: A(x) = Product_{n>=1} 1/(1 - G_n(x)^n) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x): G_n(x) = G_{n-1}(G(x)) with G_0(x)=x.

Original entry on oeis.org

1, 1, 3, 9, 31, 121, 540, 2692, 14938, 91599, 613393, 4443653, 34574511, 287092372, 2529894464, 23546940208, 230556268991, 2366688840376, 25394369705598, 284078044017592, 3305577581826931, 39928510807691616, 499751895103778173

Offset: 0

Paul D. Hanna, Dec 18 2010

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 121*x^5 + 540*x^6 +...
Let G_n(x) denote the n-th iteration of G(x) = x + x^2, then
the logarithm of A(x) begins:
log(A(x)) = G(x) + [G(x)^2 + 2*G_2(x)^2]/2 + [G(x)^3 + 3*G_3(x)^3]/3 + [G(x)^4 + 2*G_2(x)^4 + 4*G_4(x)^4]/4 + [G(x)^5 + 5*G_5(x)^5]/5 +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 81*x^4/4 + 391*x^5/5 + 2159*x^6/6 + 13049*x^7/7 + 86257*x^8/8 + 618976*x^9/9 + 4763325*x^10/10 +...
The initial iterations of G(x) = x + x^2 begin:
  G_2(x) = G(G(x)) = x + 2*x^2 + 2*x^3 + x^4;
  G_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8;
  G_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 +...;
  G_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 +...;
  G_6(x) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + 2170*x^6 +...;
See A122888 for a table of coefficients in iterations of x + x^2.
The g.f. equals the product:
A(x) = Product_{n>=1} 1/(1-G_n(x)^n) = 1/((1-x-x^2)*(1-(x+2*x^2+2*x^3+x^4)^2)*(1-(x+3*x^2+6*x^3+9*x^4+10*x^5+8*x^6+4*x^7+x^8)^3)*...)
where G_n(x) equals the n-th iteration of x+x^2.

Cf. A182970, A122888.

/* n-th Iteration of a function: */
{ITERATE(n,F,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
/* G.f.: */
{a(n)=local(F);F=exp(sum(m=1,n+1,sumdiv(m,d,d*ITERATE(d,x+x^2,n)^m/m)));polcoeff(F,n)}

{a(n)=polcoeff(1/prod(k=1,n,1-ITERATE(k,x+x^2,n)^k),n)}

G.f.: A(x) = exp( Sum_{n>=1} Sum_{d|n} d*G_d(x)^n/n ) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x).

Name changed by Paul D. Hanna, Dec 19 2010

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A194971 a(n) equals the coefficient of x^(2*n-1) in the n-th iteration of x+x^2 for n>=1.

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A194972 a(n) equals the coefficient of x^n in the (2*n-1)-th iteration of x+x^2 for n>=1.

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A182968 G.f.: A(x) = Product_{n>=1} 1/(1 - G_n(x)^n) where G(x) = x+x^2 and G_n(x) denotes the n-th iteration of G(x): G_n(x) = G_{n-1}(G(x)) with G_0(x)=x.

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