A203803 -id:A203803 - cp's OEIS frontend

A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

1, 1, 5, 85, 2928, 314925, 84974760, 63327890015, 123670531939440, 644385861467631972, 8853970669063185618000, 321538767413685546538468385, 30768712746239178236068160093280, 7755868453482819803691622493685140880, 5144106193113274410507722020733942141881664

Offset: 1

Paul D. Hanna, Jan 06 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^85 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^314925 * (1-29*x^7-x^14)^84974760 * (1-47*x^8+x^16)^63327890015 * (1-76*x^9-x^18)^123670531939440 *...).
where F(x) = exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = g.f. of A156216:
F(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
so that the logarithm of F(x) begins:
log(F(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 + 29^7*x^7/7 + 47^8*x^8/8 + 76^9*x^9/9 + 123^10*x^10/10 +...+ Lucas(n)^n*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..69

Cf. A156216, A000032 (Lucas), A203318, A203803, A203804, A203805, A203806, A203807, A203808.

a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^(#-1)&]; Array[a, 15] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^(d-1))/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^m*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n), which is the g.f. of A156216.

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

A077916 Expansion of (1-x)^(-1)/(1 + 2x - 2x^2 - x^3).

Original entry on oeis.org

1, -1, 5, -10, 30, -74, 199, -515, 1355, -3540, 9276, -24276, 63565, -166405, 435665, -1140574, 2986074, -7817630, 20466835, -53582855, 140281751, -367262376, 961505400, -2517253800, 6590256025, -17253514249, 45170286749, -118257345970, 309601751190, -810547907570

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,4,-1,-1).

Cf. A002571.
Bisections are A103433 and A103434.
Cf. A064831, A000045. A203803, A207969, A207970.

a[0] = 1; a[1] = -1; a[2] = 5; a[3] = -10; a[n_] := a[n] = -a[n-1] + 4 a[n-2] - a[n-3] - a[n-4]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
CoefficientList[Series[(1 - x)^(-1)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
Table[If[OddQ[n], (Fibonacci[2n+2]+n+1)/5, -(Fibonacci[2n+2]-n-1)/5], {n,1,20}] (* Rigoberto Florez, May 09 2019 *)

Vec((1-x)^(-1)/(1+2*x-2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Vec(1/((1-x)^2*(1+3*x+x^2)) + O(x^100)) \\ Altug Alkan, Oct 28 2015

a(n-1) = Sum_{i=1..n} (-1)^(i+1)*Fibonacci(i)*Fibonacci(i+1), n >= 1. - Alexander Adamchuk, Jun 16 2006
From R. J. Mathar, Mar 14 2011: (Start)
G.f.: 1/((1-x)^2*(1+3*x+x^2)).
a(n) = ((-1)^n*A001906(n+2)+n+2)/5. (End)
O.g.f.: exp( Sum_{n >= 1} Lucas(n)^2*(-x)^n/n ) = 1 - x + 5*x^2 - 10*x^3 + .... Cf. A203803. See also A207969 and A207970. - Peter Bala, Apr 03 2014
From Vladimir Reshetnikov, Oct 28 2015: (Start)
Recurrence (5-term): a(0) = 1, a(1) = -1, a(2) = 5, a(3) = -10, a(n) = -a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
Recurrence (4-term): a(0) = 1, a(1) = -1, a(2) = 5, n*a(n) = (1-2*n)*a(n-1) + (3*n+3)*a(n-2) + (n+1)*a(n-3).
(End)
a(n) = (F(2n+2)+n+1)/5 if n is odd and a(n)= -(F(2n+2)-n-1)/5 if n is even, where F(n) = Fibonacci numbers (A000045). - Rigoberto Florez, May 09 2019

A203804 G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 41, 126, 1526, 7854, 63629, 400789, 2870629, 19254504, 133376760, 909578760, 6249172910, 42785312510, 293403088510, 2010553849020, 13781960765020, 94458627485820, 647442212896270, 4437595353800270, 30415849505902910, 208472981440853160, 1428896115173689560

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

			G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,40,45,-285,-272,1022,370,-1840,370,1022,-272,-285,45,40,1,-1).

Cf. A002571, A203803, A203805, A203806, A203807, A203808, A203809.

Cf. A203854, A203800.

CoefficientList[Series[1/((1 - x)^6*(1 + 3*x + x^2)^4*(1 - 7*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^4*x^k/k)+x*O(x^n)), n)}

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

G.f.: 1/( (1-x)^6 * (1+3*x+x^2)^4 * (1-7*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203854(n) where A203854(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^3.

A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 365, 1730, 97390, 948562, 26292937, 370813165, 7716851405, 127699557640, 2397734250216, 42004273130216, 763345960355450, 13608990417046650, 245008471017094450, 4389301146029065420, 78826300825689660420, 1413927351334191841100, 25376664633745265522450

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

			G.f.: A(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
where
log(A(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..795
Index entries for linear recurrences with constant coefficients, order 64.

Cf. A002571, A203803, A203804, A203805, A203807, A203808, A203809.

Cf. A203856, A203800.

CoefficientList[Series[1/((1 + x)^20*(1 - 3*x + x^2)^15*(1 + 7*x + x^2)^6*(1 - 18*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^6*x^k/k)+x*O(x^n)), n)}

{a(n,m=3)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

G.f.: 1/( (1+x)^20 * (1-3*x+x^2)^15 * (1+7*x+x^2)^6 * (1-18*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203856(n) where A203856(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^5.

A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 122, 463, 11985, 85456, 1262166, 12018742, 145326748, 1540766090, 17495016342, 191731126832, 2138972609189, 23652975370501, 262682339212290, 2911255335387883, 32296421465575573, 358120616523262016, 3971885483375619384, 44047530724737577400

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
where
log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (1,121,220,-3460,-5932,52717,52667,-483925,-81600,2532240,-1172640,-6764090,4911050,11191850,-8809960,-13039640,8809960,11191850,-4911050,-6764090,1172640,2532240,81600,-483925,-52667,52717,5932,-3460,-220,121,-1,-1).

Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.

Cf. A203855, A203800.

CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}

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

G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.

A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 1094, 6555, 809765, 10676072, 570282082, 11680775298, 427757608420, 10880625876510, 341910837405634, 9500984180929624, 282684350289144641, 8100555748749977985, 236841648715969283630, 6851665210550903756723, 199305150210062939465293

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where
log(A(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..680
Index entries for linear recurrences with constant coefficients, order 128.

Cf. A002571, A203803, A203804, A203805, A203806, A203808, A203809.

Cf. A203857, A203800.

CoefficientList[Series[1/((1 + x - x^2)^35*(1 - 4*x - x^2)^21*(1 + 11*x - x^2)^7*(1 - 29*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^7*x^k/k)+x*O(x^n)), n)}

{a(n,m=3)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

G.f.: 1/( (1+x-x^2)^35 * (1-4*x-x^2)^21 * (1+11*x-x^2)^7 * (1-29*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203857(n) where A203857(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^6.

A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 3281, 25126, 6845526, 121368902, 12805025677, 373879862237, 24707348223677, 948781359159752, 50702478932197928, 2210812262034197128, 108528095366637700218, 4974402150387759436378, 236926456045384849970778, 11047772769135934828000404

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

			G.f.: A(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(A(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..596
Index entries for linear recurrences with constant coefficients, order 256.

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203809.

Cf. A203858, A203800.

CoefficientList[Series[1/((1 - x)^70*(1 + 3*x + x^2)^56*(1 - 7*x + x^2)^28*(1 + 18*x + x^2)^8*(1 - 47*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^8*x^k/k)+x*O(x^n)), n)}

{a(n,m=4)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

G.f.: 1/( (1-x)^70 * (1+3*x+x^2)^56 * (1-7*x+x^2)^28 * (1+18*x+x^2)^8 * (1-47*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203858(n) where A203858(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^7.

A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 4, 5, 10, 24, 50, 120, 270, 640, 1500, 3600, 8610, 20880, 50700, 124024, 304290, 750120, 1854400, 4600200, 11440548, 28527320, 71289000, 178526880, 447910470, 1125750120, 2833885800, 7144449920, 18036373140, 45591631800, 115381697740, 292329067800, 741410800830

Offset: 1

Paul D. Hanna, Jan 07 2012

nonn

Apparently the same as A032170, if n > 2. - R. J. Mathar, Jan 11 2012

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^4 * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^10 * (1-11*x^5-x^10)^24 * (1-18*x^6+x^12)^50 * (1-29*x^7-x^14)^120 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^3 * x^n/n ) = g.f. of A203803:
F(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 + ...
where
log(F(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 + ... + Lucas(n)^3*x^n/n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..640

Cf. A001254, A203803, A006206, A203854, A203855, A203856, A203857, A203858, A203859, A203800.

a[n_]:= 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^2 &]; Array[a, 30] (* G. C. Greubel, Dec 25 2017 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^2)/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^3*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^3 * x^n/n), which is the g.f. of A203803.

a(n) ~ phi^(2*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

A203809 G.f.: exp( Sum_{n>=1} A000204(n)^9 * x^n/n ) where A000204 is the Lucas numbers.

Original entry on oeis.org

1, 1, 9842, 97223, 58608265, 1390114224, 296390076414, 12122505505998, 1486321234837932, 84428445979241330, 7833461016478812734, 528228569507280147664, 43275470600883540869733, 3148637876123977595284117, 245565185017744596492591850

Offset: 0

Paul D. Hanna, Jan 06 2012

nonn

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

			G.f.: A(x) = 1 + x + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 + ...
where
log(A(x)) = x + 3^9*x^2/2 + 4^9*x^3/3 + 7^9*x^4/4 + 11^9*x^5/5 + 18^9*x^6/6 + 29^9*x^7/7 + 47^9*x^8/8 + ... + Lucas(n)^9*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..530
Index entries for linear recurrences with constant coefficients, order 512.

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203808.

Cf. A203859, A203800.

CoefficientList[Series[1/((1 - x - x^2)^126*(1 + 4*x - x^2)^84*(1 - 11*x - x^2)^36*(1 + 29*x - x^2)^9*(1 - 76*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^9*x^k/k)+x*O(x^n)), n)}

{a(n,m=4)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}

G.f.: 1/( (1-x-x^2)^126 * (1+4*x-x^2)^84 * (1-11*x-x^2)^36 * (1+29*x-x^2)^9 * (1-76*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203859(n) where A203859(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^8.

A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 8, 140, 1864, 26602, 373080, 5253564, 73911192, 1040045475, 14634444720, 205922568360, 2897549559600, 40771618763540, 573700205699920, 8072574516567400, 113589743388536528, 1598328982089075749, 22490195492277648120, 316461065874934143252

Offset: 0

Paul D. Hanna, May 17 2012

nonn

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).

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

			G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 + ...
where
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 + ...
Also, the g.f. equals the infinite product:
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) * ...).
The exponents in these products begin:
A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].
The companion Pell numbers begin (at offset 1):
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

G. C. Greubel, Table of n, a(n) for n = 0..865
Index entries for linear recurrences with constant coefficients, signature (8,76,136,-38,-136,76,-8,-1).

Cf. A212443, A203803, A002203, A204062.

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 *)

/* Subroutine for the PARI programs that follow: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

/* G.F. by Definition: */
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}

/* G.F. as a Finite Product: */
{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)}

/* G.F. as an Infinite Product: */
{A212443(n)=(1/n)*sumdiv(n,d, moebius(n/d)*A002203(d)^2)}
{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)}
for(n=0,30,print1(a(n),", "))

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

cp's OEIS Frontend

A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

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A077916 Expansion of (1-x)^(-1)/(1 + 2*x - 2*x^2 - x^3).

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A203804 G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.

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A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

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A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

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A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.

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A077916 Expansion of (1-x)^(-1)/(1 + 2x - 2x^2 - x^3).