A182826 -id:A182826 - cp's OEIS frontend

A182825 E.g.f. 1/(cos(sqrt(3)x) - sin(sqrt(3)x)/sqrt(3)).

1, 1, 5, 21, 153, 1209, 12285, 140589, 1871217, 27773361, 460041525, 8363802501, 166064229513, 3570030632169, 82674532955565, 2051044762727709, 54279654050034657, 1526205561241263201, 45438086217150617445, 1427921718081647393781, 47235337785416646609273

Offset: 0

Paul Barry, Dec 05 2010

nonn

First column of A182824. Hankel transform is 4^C(n+1,2)*(A000178(n))^2.

Moments of orthogonal polynomials whose coefficient array is A182826.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.

nn = 20; Table[n!, {n, 0, nn}] CoefficientList[Series[1/(Cos[Sqrt[3]*x] - Sin[Sqrt[3]*x]/Sqrt[3]), {x, 0, nn}], x] (* T. D. Noe, Jun 28 2011 *)

From Peter Bala, Jan 21 2011: (Start)
By comparing the e.g.f. for this sequence with the e.g.f for the type B Eulerian numbers A060187 we can show that
(1)... a(n) = B(n,w)/(1+w)^(n+1), where w = exp(2*Pi*I/3) and {B(n,x)}n>=1 = [x,x+x^2,x+6*x^2+x^3,x+23*x^2+23*x^3+x^4,...] are the type B Eulerian polynomials.
Equivalently,
(2)... a(n) = (-I*sqrt(3))^n*Sum_{k = 0..n} 2^k*k!*A039755(n,k)*(-1/2+sqrt(3)*I/6)^k,
where A039755(n,k) are the type B analogs of the Stirling numbers of the second kind. We can rewrite this as
(3)... a(n) = (-I*sqrt(3))^n*sum {k = 0..n} (-1/2+sqrt(3)*I/6)^k * Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n.
This explicit formula for a(n) may be used to obtain various congruence results. For example,
(4a)... a(p) = 1 (mod p) for prime p = 6*n+1,
(4b)... a(p) = -1 (mod p) for prime p = 6*n+5.
For similar results see A000111. Let u = exp(2*Pi*I/6) = 1/2+sqrt(3)/2*I be a primitive sixth root of unity.
(5)... a(n) = Sum_{k = 0..n+1} u^(n+2-2*k)*Sum_{j = 1..n+1} (-1)^(k-j)*binomial(n+1,k-j)*(2*j-1)^n. Cf. A002439. (End)
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+1) - x^2*(2*k+2)^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013
E.g.f.: 1/E(0), where E(k) = 1 - x/( 2*k+1 - 3*x*(2*k+1)/(3*x + 2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 - (1 -x -2*x*k)*(1 -3*x -2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2013

A182824 Inverse of coefficient array for orthogonal polynomials p(n,x)=(x-(2n-1))p(n-1,x)-(2n-2)^2p(n-2,x).

Original entry on oeis.org

1, 1, 1, 5, 4, 1, 21, 33, 9, 1, 153, 264, 114, 16, 1, 1209, 2769, 1410, 290, 25, 1, 12285, 32076, 20259, 5040, 615, 36, 1, 140589, 432657, 314811, 94899, 14175, 1155, 49, 1, 1871217, 6475536, 5423076, 1886304, 337974, 33936, 1988, 64, 1, 27773361, 108067041, 101497860, 40257540, 8321670, 997542, 72324, 3204, 81, 1, 460041525, 1975940244, 2064827781, 915887520, 214906770, 29709288, 2565738, 141120, 4905, 100, 1

Offset: 0

Paul Barry, Dec 05 2010

Inverse is the coefficient array for the orthogonal polynomials p(0,x)=1,p(1,x)=x-1,p(n,x)=(x-(2n-1))*p(n-1,x)-(2n-2)^2*p(n-2,x).

Inverse is A182826. First column is A182825.

			Triangle begins:
  1,
  1, 1,
  5, 4, 1,
  21, 33, 9, 1,
  153, 264, 114, 16, 1,
  1209, 2769, 1410, 290, 25, 1,
  12285, 32076, 20259, 5040, 615, 36, 1,
  140589, 432657, 314811, 94899, 14175, 1155, 49, 1,
  1871217, 6475536, 5423076, 1886304, 337974, 33936, 1988, 64, 1
Production matrix begins:
  1, 1,
  4, 3, 1,
  0, 16, 5, 1,
  0, 0, 36, 7, 1,
  0, 0, 0, 64, 9, 1,
  0, 0, 0, 0, 100, 11, 1,
  0, 0, 0, 0, 0, 144, 13, 1,
  0, 0, 0, 0, 0, 0, 196, 15, 1,
  0, 0, 0, 0, 0, 0, 0, 256, 17, 1
  0, 0, 0, 0, 0, 0, 0, 0, 324, 19, 1

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(Cos[Sqrt[3]*#] - Sin[Sqrt[3]*#]/Sqrt[3])&, Sin[Sqrt[3]*#]/ (Sqrt[3]*Cos[Sqrt[3]*#] - Sin[Sqrt[3]*#])&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Exponential Riordan array [1/(cos(sqrt(3)*x)-sin(sqrt(3)*x)/sqrt(3)), sin(sqrt(3)*x)/(sqrt(3)*cos(sqrt(3)*x)-sin(sqrt(3)*x))].

A182827 E.g.f. 1/sqrt(1+2x+4x^2).

Original entry on oeis.org

1, -1, -1, 21, -111, -345, 14895, -143955, -760095, 49774095, -699437025, -5221460475, 458621111025, -8457966542025, -81662774418225, 8999266227076125, -205480756062957375, -2434383666448358625, 322739182334471277375, -8786388514658364484875

Offset: 0

Paul Barry, Dec 05 2010

First column of A182826.

Table[(-1)^n*2^n*n!*LegendreP[n, 1/2], {n, 0, 20}] (* Vaclav Kotesovec, May 06 2017 *)

x='x+O('x^55); Vec(serlaplace(1/sqrt(1+2*x+4*x^2)))

D-finite with recurrence: a(n) +(2n-1)*a(n-1) +4*(n-1)^2*a(n-2)=0. - R. J. Mathar, Nov 17 2011

cp's OEIS Frontend

A182825 E.g.f. 1/(cos(sqrt(3)x) - sin(sqrt(3)x)/sqrt(3)).

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A182824 Inverse of coefficient array for orthogonal polynomials p(n,x)=(x-(2n-1))p(n-1,x)-(2n-2)^2p(n-2,x).

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A182827 E.g.f. 1/sqrt(1+2x+4x^2).

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cp's OEIS Frontend

A182825 E.g.f. 1/(cos(sqrt(3)*x) - sin(sqrt(3)*x)/sqrt(3)).

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A182824 Inverse of coefficient array for orthogonal polynomials p(n,x)=(x-(2n-1))*p(n-1,x)-(2n-2)^2*p(n-2,x).

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A182827 E.g.f. 1/sqrt(1+2x+4x^2).

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Author

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Mathematica

PARI

Formula

A182825 E.g.f. 1/(cos(sqrt(3)x) - sin(sqrt(3)x)/sqrt(3)).

A182824 Inverse of coefficient array for orthogonal polynomials p(n,x)=(x-(2n-1))p(n-1,x)-(2n-2)^2p(n-2,x).