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 1-5 of 5 results.

A203860 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -1, -4, -1, 1, 11, 7, 25, 18, -11, -1, 0, -325, -199, 122, -1364, -843, 550, 0, 11, 123, 0, 39650, 24476, -15126, 0, 271443, 164194, -103682, -1364, -1, -24476, 0, -9349, -123, -20633239, -12752043, 7881225, -843, 0, -226965629, -141422125, 88114450, 0, 1
Offset: 0

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Author

Paul D. Hanna, Jan 07 2012

Keywords

Comments

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

Examples

			G.f.: A(x) = 1 - x - 4*x^2 - x^3 + x^4 + 11*x^5 + 7*x^6 + 25*x^7 +...
-log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...]
which are numbers that are not the sum of two generalized pentagonal numbers.

Links

Crossrefs

Cf. A203861, A203850, A156234, A093519.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

Formula

G.f.: exp( Sum_{n>=1} -sigma(n) * A000204(n) * x^n/n ).

A203861 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -3, -9, 20, 45, 0, -151, -231, 0, 140, 1107, 2052, 49, -1305, 0, -15004, -28260, 0, 17710, 0, 81, 324040, 589953, 0, -375570, -1089, 0, -124124, -10659705, -19764180, -121, 12605358, 0, 0, 4158315, 0, 567552368, 1052295189, -780030, -669901660, 0, 0, -221399431, -85965, 0
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Comments

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

Examples

			G.f.: A(x) = 1 - 3*x - 9*x^2 + 20*x^3 + 45*x^4 - 151*x^6 - 231*x^7 +...
-log(A(x))/3 = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2)^3 * (1-3*x^2+x^4)^3 * (1-4*x^3-x^6)^3 * (1-7*x^4+x^8)^3 * (1-11*x^5-x^10)^3 * (1-18*x^6+x^12)^3 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...]
which are numbers that are not the sum of two triangular numbers.

Links

Crossrefs

Cf. A203860, A203850, A156234, A020757.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, -3*sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^3, n)}

Formula

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A000204(n) * x^n/n ).

A225528 a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

Original entry on oeis.org

1, 9, 16, 49, 66, 216, 232, 705, 988, 2214, 2388, 9016, 7294, 20232, 32736, 68417, 64278, 225342, 186980, 635334, 783232, 1425708, 1537896, 6220920, 5200591, 11400606, 17568160, 39796232, 34495530, 133955856, 96331168, 306863361, 378297408, 688610322, 990395472, 3038060662
Offset: 1

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Author

Paul D. Hanna, May 09 2013

Keywords

Examples

			L.g.f.: L(x) = x + 9*x^2/2 + 16*x^3/3 + 49*x^4/4 + 66*x^5/5 + 216*x^6/6 +...
which is equivalent to:
L(x) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + 8*29*x^7/7 + 15*47*x^8/8 +...+ sigma(n)*Lucas(n)*x^n/n +...
where exponentiation yields the g.f. of A156234:
exp(L(x)) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 +...+ A156234(n)*x^n +...
and equals the product:
exp(L(x)) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(n)*x^n + (-x^2)^n) *...).

Links

Crossrefs

Cf. A156234, A225525.

Programs

  • PARI
    {a(n)=sigma(n)*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1,40,print1(a(n),", "))
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1, n, -log(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=1,40,print1(a(n),", "))

Formula

L.g.f.: Sum_{n>=1} -log(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} a(n)*x^n/n.
Logarithmic derivative of A156234.

A203801 G.f.: Product_{n>=1} (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 1, 2, 7, 9, 27, 53, 109, 206, 463, 907, 1756, 3591, 6849, 13706, 27132, 51477, 99168, 195160, 366269, 707173, 1355524, 2558372, 4836092, 9186600, 17245564, 32428375, 61057276, 113946770, 212495896, 397836811, 737325660, 1368659832, 2544085015, 4694930535
Offset: 0

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Author

Paul D. Hanna, Jan 11 2012

Keywords

Comments

Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions into distinct parts.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 9*x^4 + 27*x^5 + 53*x^6 +...
where A(x) = (1+x-x^2) * (1+3*x^2+x^4) * (1+4*x^3-x^6) * (1+7*x^4+x^8) * (1+11*x^5-x^10) * (1+18*x^6+x^12) *...* (1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) *...
and 1/A(x) = (1-x-x^2) * (1-4*x^3-x^6) * (1-11*x^5-x^10) * (1-29*x^7-x^14) * (1-76*x^9-x^18) * (1-199*x^11-x^22) *...* (1 - Lucas(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...
Also, the logarithm of the g.f. equals the series:
log(A(x)) = x + 1*3*x^2/2 + 4*4*x^3/3 + 1*7*x^4/4 + 6*11*x^5/5 + 4*18*x^6/6 + 8*29*x^7/7 + 1*47*x^8/8 +...+ A000593(n)*Lucas(n)*x^n/n +...

Links

Crossrefs

Cf. A156234, A203860, A203850, A000204, A000593.

Programs

  • Mathematica
    max = 40; s = Product[1 + LucasL[n]*x^n + (-1)^n*x^(2*n), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 14 2015 *)
  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(prod(k=1,n,1+Lucas(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}
  • PARI
    {a(n)=polcoeff(1/prod(k=1,n,1-Lucas(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)),n)}
  • PARI
    /* Exponential form using sum of odd divisors of n: */
{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}

Formula

G.f.: Product_{n>=1} 1/(1 - Lucas(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).
G.f.: exp( Sum_{n>=1} A000593(n) * Lucas(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.
a(n) = (1/n)*Sum_{k=1..n} A000593(k)*Lucas(k)*a(n-k) for n>0, with a(0) = 1.

A203534 G.f.: exp( Sum_{n>=1} sigma(n)*A002203(n)*x^n/n ) where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 2, 11, 38, 156, 516, 1991, 6434, 23065, 75132, 255335, 816480, 2724245, 8570794, 27763807, 87057596, 276339126, 855374534, 2681503010, 8218321006, 25421912010, 77383062314, 236519199902, 714226056554, 2165295121179, 6490447624984, 19503550719297, 58127246438024
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2012

Keywords

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), and to the g.f. of Pell numbers: exp( Sum_{n>=1} A002203(n)*x^n/n ).

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 38*x^3 + 156*x^4 + 516*x^5 + 1991*x^6 +...
where
A(x) = 1/((1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...).
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...].

Crossrefs

Cf. A156234, A000129 (Pell), A002203 (companion Pell), A000203 (sigma).

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
  • PARI
    {a(n)=if(n<0,0,if(n==0,1,(1/n)*sum(k=1,n,sigma(k)*A002203(k)*a(n-k))))}

Formula

G.f.: Product_{n>=1} 1/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)).
a(n) = (1/n)*Sum_{k=1..n} sigma(k)*A002203(k)*a(n-k) for n>0, with a(0) = 1.
Showing 1-5 of 5 results.

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