A212442 - OEIS

%I #17 Dec 26 2017 03:16:15

%S 1,8,140,1864,26602,373080,5253564,73911192,1040045475,14634444720,

%T 205922568360,2897549559600,40771618763540,573700205699920,

%U 8072574516567400,113589743388536528,1598328982089075749,22490195492277648120,316461065874934143252

%N G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

%C More generally, exp(Sum_{k>=1} A002203(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A002203(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

%C Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

%H G. C. Greubel, <a href="/A212442/b212442.txt">Table of n, a(n) for n = 0..865</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,76,136,-38,-136,76,-8,-1).

%F G.f.: 1 / ( (1+2*x-x^2)^3 * (1-14*x-x^2) ).

%F G.f.: 1 / Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) where A212443(n) = (1/n)*Sum_{d|n} moebius(n/d)*A002203(d)^2.

%F a(0)=1, a(1)=8, a(2)=140, a(3)=1864, a(4)=26602, a(5)=373080, a(6)=5253564, a(7)=73911192, a(n) = 8*a(n-1) + 76*a(n-2) + 136*a(n-3) - 38*a(n-4) - 136*a(n-5) + 76*a(n-6) - 8*a(n-7) - a(n-8). - _Harvey P. Dale_, Feb 15 2015

%e G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 + ...

%e where

%e log(A(x)) = 2^3*x + 6^3*x^2/2 + 14^3*x^3/3 + 34^3*x^4/4 + 82^3*x^5/5 + 198^3*x^6/6 + 478^3*x^7/7 + 1154^3*x^8/8 + ... + A002203(n)^3*x^n/n + ...

%e Also, the g.f. equals the infinite product:

%e A(x) = 1/( (1-2*x-x^2)^4 * (1-6*x^2+x^4)^16 * (1-14*x^3-x^6)^64 * (1-34*x^4+x^8)^280 * (1-82*x^5-x^10)^1344 * (1-198*x^6+x^12)^6496 * ... * (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) * ...).

%e The exponents in these products begin:

%e A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].

%e The companion Pell numbers begin (at offset 1):

%e A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

%t CoefficientList[Series[1/((1+2x-x^2)^3(1-14x-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{8,76,136,-38,-136,76,-8,-1},{1,8,140,1864,26602,373080,5253564,73911192},30] (* _Harvey P. Dale_, Feb 15 2015 *)

%o (PARI) /* Subroutine for the PARI programs that follow: */

%o {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

%o (PARI) /* G.F. by Definition: */

%o {a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}

%o (PARI) /* G.F. as a Finite Product: */

%o {a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*A002203(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}

%o (PARI) /* G.F. as an Infinite Product: */

%o {A212443(n)=(1/n)*sumdiv(n,d, moebius(n/d)*A002203(d)^2)}

%o {a(n)=polcoeff(1/prod(m=1,n, (1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^A212443(m)),n)}

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

%Y Cf. A212443, A203803, A002203, A204062.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 17 2012