A203413 G.f.: exp( Sum_{n>=1} A203414(n)*x^n/n ) where A203414(n) = n*Pell(n)*Sum_{d|n} 1/(d*Pell(d)).

1, 1, 3, 8, 25, 64, 200, 512, 1528, 4048, 11654, 30585, 88601, 231295, 651713, 1733011, 4814031, 12685230, 35225415, 92628772, 254268558, 672643614, 1826716115, 4814931851, 13086575526, 34391797265, 92637759753, 244294085952, 654813738224, 1720509596070, 4606408076053

Offset: 0

Author

Paul D. Hanna, Jan 01 2012

Keywords

• nonn

Comments

Note: x/(1-2*x-x^2) = exp(Sum_{n>=1} A002203(n)*x^n/n) is the g.f. of the Pell numbers and A002203 is the companion Pell numbers.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 64*x^5 + 200*x^6 + 512*x^7 +...
where
log(A(x)) = x/(1-2*x-x^2) + (x^2/2)/(1-6*x^2+x^4) + (x^3/3)/(1-14*x^3-x^6) + (x^4/4)/(1-34*x^4+x^8) +...+ (x^n/n)/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
Equivalently, log(A(x)) = Sum_{n>=1} G_n(x^n) * x^n/n
where G_n(x) = exp( Sum_{k>=1} A002203(n*k)*x^k/k ), which begin:
G_1(x) = x*(1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 +...+ Pell(n+1)*x^n +...
G_2(x) = 1 + 6*x^2 + 35*x^4 + 204*x^6 +...+ Pell(2*n+2)/2*x^(2*n) +...
G_3(x) = 1 + 14*x^3 + 197*x^6 + 2772*x^9 +...+ Pell(3*n+3)/5*x^(3*n) +...
G_4(x) = 1 + 34*x^4 + 1155*x^8 + 39236*x^12 +...+ Pell(4*n+4)/12*x^(4*n) +...
G_5(x) = 1 + 82*x^5 + 6725*x^10 + 551532*x^15 +...+ Pell(5*n+5)/29*x^(5*n) +...
G_6(x) = 1 + 198*x^6 + 39203*x^12 + 7761996*x^18 +...+ Pell(6*n+6)/70*x^(6*n) +...
For n>=1, G_n(x) = 1/(1 - A002203(n)*x + (-1)^n*x^2),
where the companion Pell numbers (offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726 16238,...].
The logarithmic derivative of this sequence begins:
A203414 = [1,5,16,61,146,554,1184,4149,9457,29890,63152,...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A203413, A203319, A203321; A000129 (Pell), A002203 (companion Pell).

Programs

PARI

/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}

PARI

{a(n)=local(A=1);A=exp(sum(m=1,n+1,x^m*Pell(m)*sumdiv(m, d, 1/(d*Pell(d))) +x*O(x^n)));polcoeff(A,n)}

PARI

{a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))));polcoeff(A,n)}

PARI

{a(n)=local(A=1);A=exp(sum(m=1,n+1,(x^m/m)*exp(sum(k=1,floor((n+1)/m),A002203(m*k)*x^(m*k)/k)+x*O(x^n))));polcoeff(A, n)}

PARI

{a(n)=local(A=1+2*x+x*O(x^n),G=1/(1-2*x-x^2+x*O(x^n)));A=exp(sum(m=1,n+1,(x^m/m)*round(prod(k=0,m-1,subst(G,x,exp(2*Pi*I*k/m)*x+x*O(x^n))))));polcoeff(A, n)}

Formula

G.f.: exp( Sum_{n>=1} (x^n/n) / (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) ).

G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} A002203(n*k)*x^(n*k)/k ) ).

G.f.: exp( Sum_{n>=1} G_n(x^n) * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-2*x-x^2) and u is an n-th root of unity.