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.

A084769 a(n) = P_n(9), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 9*x + 20*x^2)^n.

Original entry on oeis.org

1, 9, 121, 1809, 28401, 458649, 7544041, 125700129, 2114588641, 35836273449, 610897146201, 10463745263409, 179939616743121, 3104680678772409, 53721299280288201, 931852905510160449, 16198821321758152641
Offset: 0

Views

Author

Paul D. Hanna, Jun 03 2003

Keywords

Links

Crossrefs

Column k=4 of A335333.
Sequences of the form LegendreP(n, 2*m+1): A000012 (m=0), A001850 (m=1), A006442 (m=2), A084768 (m=3), this sequence (m=4).
Cf. A243007 (a(n)^2), A269732.

Programs

  • Magma
    [Evaluate(LegendrePolynomial(n), 9) : n in [0..40]]; // G. C. Greubel, May 17 2023
  • Mathematica
    Table[SeriesCoefficient[1/Sqrt[1-18*x+x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
LegendreP[Range[0, 40], 9] (* G. C. Greubel, May 17 2023 *)
a[n_] := 4^n*Sum[(5/4)^k*Binomial[n, k]^2, {k, 0, n}];
Table[a[n], {n, 0, 16}]  (* Detlef Meya, May 22 2024 *)
  • PARI
    for(n=0,30,print1(subst(pollegendre(n),x,9)","))
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)^2*4^k*5^(n-k))} \\ Paul D. Hanna, Sep 29 2012
  • SageMath
    [gen_legendre_P(n,0,9) for n in range(41)] # G. C. Greubel, May 17 2023

Formula

G.f.: 1/sqrt(1-18*x+x^2).
Also a(n) = (n+1)-th term of the binomial transform of 1/(1-4x)^(n+1).
From Paul Barry, May 25 2005: (Start)
E.g.f.: exp(9*x) * Bessel_I(0, 2*sqrt(20)*x).
a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)4^k. (End)
D-finite with recurrence: n*a(n) + 9*(1-2*n)*a(n-1) + (n-1)*a(n-2) = 0. - R. J. Mathar, Sep 27 2012
a(n) = Sum_{k=0..n} binomial(n,k)^2 * 4^k * 5^(n-k). - Paul D. Hanna, Sep 29 2012
a(n) ~ sqrt(200 + 90*sqrt(5))*(9 + 4*sqrt(5))^n/(20*sqrt(Pi*n)) = (2 + sqrt(5))^(2*n+1)/(5^(1/4)*2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = hypergeom([-n, n+1], [1], -4). - Peter Luschny, May 23 2014
x*exp(Sum_{n >= 1} a(n)*x^n/n) = x + 9*x^2 + 101*x^3 + 1269*x^4 + ... is an integral power series, the o.g.f. for A269732. - Peter Bala, Jan 25 2018
a(n) = (-1)^n * Sum_{k=0..n} (1/18)^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 28 2025
a(n) = Sum_{k=0..floor(n/2)} 20^k * 9^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, Aug 30 2025

A269730 Dimensions of the 2-polytridendriform operad TDendr_2.

Original entry on oeis.org

1, 5, 31, 215, 1597, 12425, 99955, 824675, 6939769, 59334605, 513972967, 4501041935, 39784038517, 354455513105, 3179928556219, 28701561707675, 260447708523505, 2374690737067925, 21744508765633327, 199877846477679815, 1843718766426242221, 17060955558786455705, 158333204443000060291
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2016

Keywords

Links

Crossrefs

Cf. A001047, A001263, A126216, A269731, A269732, A001003, A006442.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(1 - 5*x - Sqrt[1 - 10*x + x^2])/(12*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 5]/Sqrt[6], {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
  • PARI
    A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=2); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016
  • PARI
    my(q=2, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

Formula

a(n) = P_n(2), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(12 + 5*sqrt(6)) * (5 + 2*sqrt(6))^n / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A001047(x))(-x). - Gheorghe Coserea, Sep 30 2017
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 2.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 2. (End)
G.f. A(x) = x*exp( Sum_{n >= 1} A006442(n)*x^n/n ). - Peter Bala, Jan 09 2022

Extensions

More terms from Gheorghe Coserea, Apr 23 2016

A269731 Dimensions of the 3-polytridendriform operad TDendr_3.

Original entry on oeis.org

1, 7, 61, 595, 6217, 68047, 770149, 8939707, 105843409, 1273241431, 15517824973, 191202877411, 2377843390873, 29807864423071, 376255282112629, 4778240359795147, 61007205215610529, 782648075371992487, 10083436451634033757, 130413832663780730995, 1692599303723819234281, 22037570163808433691247, 287762084009227350367621
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2016

Keywords

Links

Crossrefs

Cf. A001003, A005061, A001263, A126216, A269730, A269732.

Programs

  • Magma
    I:=[1,7]; [n le 2 select I[n] else (7*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2016
  • Mathematica
    Rest[CoefficientList[Series[(1 - 7*x - Sqrt[1 - 14*x + x^2])/(24*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 7]/(2*Sqrt[3]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 7, (n + 1) a[n] == 7 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Nov 29 2016 *)
  • PARI
    A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=3); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016
  • PARI
    my(q=3, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

Formula

a(n) = P_n(3), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(24 + 14*sqrt(3)) * (7 + 4*sqrt(3))^n / (24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A005061(x))(-x). - Gheorghe Coserea, Sep 30 2017
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 3.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 3. (End)

Extensions

More terms from Gheorghe Coserea, Apr 23 2016
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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