cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A203318 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n).

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 36, 64, 135, 250, 504, 917, 1864, 3372, 6593, 12176, 23473, 42732, 82142, 149282, 283104, 516780, 967894, 1757865, 3291964, 5959633, 11039163, 20022457, 36908442, 66637739, 122512809, 220717328, 403499293, 726866565, 1322670966, 2376541137
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2011

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} F_n(x^n) * x^n/n )
where F_n(x) = exp( Sum_{k>=1} Lucas(n*k)*x^k/k ), which begin:
F_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 +...;
F_2(x) = 1 + 3*x + 8*x^2 + 21*x^3 + 55*x^4 + 144*x^5 + 377*x^6 +...;
F_3(x) = 1 + 4*x + 17*x^2 + 72*x^3 + 305*x^4 + 1292*x^5 + 5473*x^6 +...;
F_4(x) = 1 + 7*x + 48*x^2 + 329*x^3 + 2255*x^4 + 15456*x^5 +...;
F_5(x) = 1 + 11*x + 122*x^2 + 1353*x^3 + 15005*x^4 + 166408*x^5 +...;
F_6(x) = 1 + 18*x + 323*x^2 + 5796*x^3 + 104005*x^4 + 1866294*x^5 +...;
...
Also, F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where u = n-th root of unity:
F_1(x) = F(x) = 1/(1-x-x^2) = g.f. of the Fibonacci numbers;
F_2(x^2) = F(x)*F(-x) = 1/(1-3*x^2+x^4);
F_3(x^3) = F(x)*F(w*x)*F(w^2*x) = 1/(1-4*x^3-x^6) where w = exp(2*Pi*I/3);
F_4(x^4) = F(x)*F(I*x)*F(-x)*F(-I*x) = 1/(1-7*x^4+x^8);
F_5(x^5) = 1/(1-11*x^5-x^10);
In general,
F_n(x^n) = 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
...
The logarithmic derivative of this sequence begins:
A203319 = [1,3,7,19,26,81,92,267,358,848,980,3061,3030,7976,...].

Links

Crossrefs

Cf. A203319, A203320, A000032 (Lucas); A203413 (Pell variant).

Programs

  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(exp(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n)))),n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=vector(n+1, i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}

Formula

G.f.: exp( Sum_{n>=1} A203319(n)*x^n/n ) where A203319(n) = n*fibonacci(n)*Sum_{d|n} 1/(d*fibonacci(d)).
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) ) where Lucas(n) = A000032(n).
G.f.: exp( Sum_{n>=1} F_n(x^n) * x^n/n ) such that F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where F(x) = 1/(1-x-x^2) and u is an n-th root of unity.

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)).

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Jan 01 2012

Keywords

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

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.

A203414 a(n) = n*Pell(n) * Sum_{d|n} 1/(d*Pell(d)) where Pell(n) = A000129(n).

Original entry on oeis.org

1, 5, 16, 61, 146, 554, 1184, 4149, 9457, 29890, 63152, 222850, 434994, 1414642, 3140576, 9575893, 19323714, 65160959, 125877072, 408744626, 865638272, 2563647322, 5176349664, 17476326546, 33019614771, 102921708050, 220209942688, 657218691722, 1292253982322
Offset: 1

Views

Author

Paul D. Hanna, Jan 01 2012

Keywords

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 61*x^4/4 + 146*x^5/5 + 554*x^6/6 +...
where
L(x) = x*(1 + 2*x + 5*x^2 + 12*x^3 + 29*x^4 +...+ Pell(n+1)*x^n +...) +
x^2/2*(1 + 6*x^2 + 35*x^4 + 204*x^6 +...+ Pell(2*n+2)/2*x^(2*n) +...) +
x^3/3*(1 + 14*x^3 + 197*x^6 + 2772*x^9 +...+ Pell(3*n+3)/5*x^(3*n) +...) +
x^4/4*(1 + 34*x^4 + 1155*x^8 + 39236*x^12 +...+ Pell(4*n+4)/12*x^(4*n) +...) +
x^5/5*(1 + 82*x^5 + 6725*x^10 + 551532*x^15 +...+ Pell(5*n+5)/29*x^(5*n) +...) +
x^6/6*(1 + 198*x^6 + 39203*x^12 + 7761996*x^18 +...+ Pell(6*n+6)/70*x^(6*n) +...) +...
Equivalently,
L(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)) +...
where A002203 is the companion Pell numbers.
Exponentiation of the l.g.f. equals the g.f. of A203413:
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 64*x^5 + 200*x^6 + 512*x^7 + 1528*x^8 + 4048*x^9 +...+ A203413(n)*x^n +...

Links

Crossrefs

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

Programs

  • Mathematica
    pell[n_] := pell[n] = ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2]) // Round; a[n_] := n pell[n] DivisorSum[n, 1/(# pell[#]) &]; Array[a, 30] (* Jean-François Alcover, Dec 23 2015 *)
  • 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)=if(n<1, 0, n*Pell(n)*sumdiv(n, d, 1/(d*Pell(d))) )}
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n+1, (x^m/m)/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
  • PARI
    {a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), A002203(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L, n)}
  • PARI
    {a(n)=local(A=1+2*x+x*O(x^n), F=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(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Formula

Equals the logarithmic derivative of A203413.
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} Pell(n*k+n)/Pell(n) * x^(n*k).
(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} A002203(n*k)*x^(n*k)/k ).
(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)).
(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where 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.
Showing 1-3 of 3 results.

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