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 1-2 of 2 results.

A099325 Expansion of (sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(3/2)).

Original entry on oeis.org

1, 3, 7, 16, 35, 76, 162, 344, 723, 1516, 3158, 6568, 13598, 28120, 57956, 119344, 245123, 503116, 1030542, 2109704, 4311786, 8808328, 17969372, 36644176, 74640430, 151985016, 309170332, 628741264, 1277540828, 2595198256
Offset: 0

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Author

Paul Barry, Oct 12 2004

Keywords

Comments

The g.f. is transformed to 1/(1-x)^3 under the Chebyshev transformation A(x)->1/(1+x^2)A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x), where c(x) is the g.f. of the Catalan numbers A000108.
Image of 2n+1 under the Riordan array (1/sqrt(1-4x^2),xc(x^2)). Hankel transform is (n+1)*(-1)^n. - Paul Barry, Oct 06 2007
a(n) is the minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n+1, and each number in a higher row is the sum of the two numbers directly below it. - Nathaniel Johnston, Apr 20 2011

Links

Crossrefs

Cf. A066411, A099326, A099327.

Programs

  • Maple
    a:=proc(n)return 2^n+add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end:
seq(a(n), n=0..30); # Nathaniel Johnston, Apr 20 2011
  • Mathematica
    CoefficientList[Series[(Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(3/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+2, 2)*(1+(-1)^(n-k))/(n+k+2).
a(n) = 2^n + Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n, k). - Nathaniel Johnston, Apr 20 2011
a(n) = M^n*V topmost term. M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals, [1,0,0,0,...] as the main diagonal; and the rest zeros. V = the vector [1,2,3,...]. - Gary W. Adamson, Jan 30 2012
D-finite with recurrence n*a(n) + 2*(-n-1)*a(n-1) + 4*(-n+3)*a(n-2) + 8*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ 2^(n+1/2)*sqrt(n)/sqrt(Pi) * (1 + sqrt(2*Pi/n)/4). - Vaclav Kotesovec, Feb 13 2014

A099326 Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).

Original entry on oeis.org

1, 4, 11, 28, 67, 156, 354, 792, 1747, 3820, 8278, 17832, 38174, 81368, 172644, 365104, 769411, 1617228, 3389838, 7090440, 14797546, 30828424, 64106716, 133113168, 275967022, 571415416, 1181585564, 2440680592, 5035637212
Offset: 0

Views

Author

Paul Barry, Oct 12 2004

Keywords

Comments

a(n) = Sum_{k=0..n} (k+1)*binomial(n,(n-k)/2)*binomial(k+3,3)*(1+(-1)^(n-k))/(n+k+2). The g.f. is transformed to 1/(1-x)^4 under the Chebyshev transformation A(x) -> (1/(1+x^2))*A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x)^2, where c(x) is the g.f. of the Catalan numbers A000108.
0, 1, 4, 11, 28, ... is the image of the quarter-squares floor((n+1)^2/4) (A002620(n+1)) under the Riordan array ((1+2x)/sqrt(1-4x^2), x*c(x^2)). Hankel transform of A099326 has g.f. (1-x)/(1+x)^4. - Paul Barry, Oct 25 2007

Links

Crossrefs

Cf. A099325, A099327.

Programs

  • Mathematica
    CoefficientList[Series[((1-2*x)*Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Formula

a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+3, 3)*(1 + (-1)^(n-k))/(n+k+2).
a(n) = Sum_{k=0..n} C(n,k)*(floor((abs(n-2k) + 1)^2/4) + floor((abs(n-2k+1) + 1)^2/4)). - Paul Barry, Oct 25 2007
D-finite with recurrence: n*(n-2)*a(n) +2*(-n^2+3)*a(n-1) -4*(n-1)*(n-4)*a(n-2) +8*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ n * 2^(n-1) * (1 + 2*sqrt(2/(Pi*n))). - Vaclav Kotesovec, Feb 12 2014
Showing 1-2 of 2 results.

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