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.

A322243 a(n) = A322242(n)^2, the square of the central coefficient in (1 + 3*x + 4x^2)^n.

Original entry on oeis.org

1, 9, 289, 9801, 370881, 14768649, 609546721, 25795893321, 1112301387649, 48661046208009, 2153525838773409, 96206353829163081, 4331637064535243841, 196320612369490652169, 8948100956521251936609, 409841929215895450531401, 18851871634769751620818689, 870412126485504031282344969, 40322236319029726815932366881, 1873525179289326411511891685961, 87284881539374630658475997415361
Offset: 0

Views

Author

Paul D. Hanna, Dec 08 2018

Keywords

Comments

The g.f. of A322242 is 1/sqrt(1 - 6*x - 7*x^2).

Examples

			G.f.: A(x) = 1 + 9*x + 289*x^2 + 9801*x^3 + 370881*x^4 + 14768649*x^5 + 609546721*x^6 + 25795893321*x^7 + 1112301387649*x^8 + 48661046208009*x^9 + ...
such that
A(x) = 1 + 3^2*x + 17^2*x^2 + 99^2*x^3 + 609^2*x^4 + 3843^2*x^5 + 24689^2*x^6 + 160611^2*x^7 + 1054657^2*x^8 + 6975747^2*x^9 + ... + A322242(n)^2*x^n + ...

Links

Crossrefs

Cf. A322242.

Programs

  • Maple
    f:= gfun:-rectoproc({343*(5+2*n)*(n+1)^2*a(n)-7*(3+2*n)*(43*n^2+172*n+163)*a(n+1)-(5+2*n)*(43*n^2+172*n+163)*a(n+2)+(3+2*n)*(n+3)^2*a(n+3)=0, a(0)=1, a(1)=3^2, a(2)=17^2},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 10 2018
  • Mathematica
    f[n_] := (CoefficientList[Expand[(1 + 3*x + 4*x^2)^n], x][[n + 1]])^2; Array[f, 22, 0] (* Amiram Eldar, Dec 10 2018 *)
CoefficientList[Series[2*EllipticK[1 - (1 + 7*x)^2/((1 - 49*x)*(1 - x))] / (Pi*Sqrt[(1 - 49*x)*(1 - x)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 27 2019 *)
  • PARI
    /* a(n) = A322242(n)^2 */
{a(n)=polcoeff(1/sqrt(1 - 6*x - 7*x^2 +x*O(x^n)), n)^2}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Using AGM: */
{a(n)=polcoeff( 1 / 1 / agm(1 + 7*x, sqrt((1 - x)*(1 - 7^2*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: 1 / AGM(1 + 7*x, sqrt((1 - x)*(1 - 49*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-x)*(1-7*x), (1+x)*(1+7*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) = A322242(n)^2 where A322242(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).
343*(5+2*n)*(n+1)^2*a(n)-7*(3+2*n)*(43*n^2+172*n+163)*a(n+1)-(5+2*n)*(43*n^2+172*n+163)*a(n+2)+(3+2*n)*(n+3)^2*a(n+3)=0. - Robert Israel, Dec 10 2018
a(n) ~ 7^(2*n + 1) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2019

Keywords

Comments

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Links

Crossrefs

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A383944 Expansion of 1/sqrt((1-7*x)^3 * (1+x)).

Original entry on oeis.org

1, 10, 87, 708, 5565, 42798, 324275, 2430536, 18068409, 133454610, 980588367, 7174290060, 52301288949, 380120468406, 2755437681195, 19928252747664, 143839643441265, 1036380251867418, 7455465737930567, 53557027924956500, 384241833300244269, 2753539115904779070
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A322242, A383945.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1-7*x)^3 * (1+x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    CoefficientList[Series[1/Sqrt[(1-7*x)^3*(1+x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-7*x)^3*(1+x)))

Formula

n*a(n) = (6*n+4)*a(n-1) + 7*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

A383945 Expansion of 1/sqrt((1-7*x)^5 * (1+x)).

Original entry on oeis.org

1, 17, 206, 2150, 20615, 187103, 1633996, 13868508, 115147965, 939490365, 7557020922, 60073436514, 472815344547, 3689827880235, 28584232842840, 220017882647544, 1683964821974073, 12824134005685929, 97224403777732070, 734127854369080990, 5523136813883811199
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A322242, A383944.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1-7*x)^5 * (1+x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    CoefficientList[Series[1/Sqrt[(1-7*x)^5*(1+x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-7*x)^5*(1+x)))

Formula

n*a(n) = (6*n+11)*a(n-1) + 7*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+2,n-k).

A383947 Expansion of 1/sqrt((1+x)^3 * (1-7*x)).

Original entry on oeis.org

1, 2, 15, 84, 525, 3318, 21371, 139240, 915417, 6060330, 40345767, 269825724, 1811432805, 12200012958, 82394389395, 557794589904, 3784079617713, 25718668160850, 175085306697791, 1193682452744740, 8148955372804029, 55697327430265158, 381099865385716395
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A085458, A322242.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1+x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    CoefficientList[Series[1/Sqrt[(1+x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt((1+x)^3*(1-7*x)))

Formula

n*a(n) = (6*n-4)*a(n-1) + 7*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).
Showing 1-5 of 5 results.

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