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.

A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 25, 1, 5, 43, 425, 1, 7, 65, 661, 7025, 1, 9, 91, 965, 10515, 116625, 1, 11, 121, 1345, 15105, 171097, 1951625, 1, 13, 155, 1809, 20995, 243525, 2828101, 32903225, 1, 15, 193, 2365, 28401, 337877, 4001345, 47284251, 558265825
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Examples

			[0] 1,  1,  25,  425,  7025, 116625,  1951625,  32903225, ... [A299845]
[1] 1,  3,  43,  661, 10515, 171097,  2828101,  47284251, ... [A299506]
[2] 1,  5,  65,  965, 15105, 243525,  4001345,  66622085, ...
[3] 1,  7,  91, 1345, 20995, 337877,  5544709,  92234527, ... [A243946]
[4] 1,  9, 121, 1809, 28401, 458649,  7544041, 125700129, ... [A084769]
[5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947]
[6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...

Crossrefs

Cf. A299845, A299506, A243946, A084769, A243947.
Cf. A300946.

Programs

  • Mathematica
    Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}];
Table[Print[Arow[n, 7]], {n, 0, 6}];
T[n_, k_] := If[k==0, 1, 4^k*Sum[(5/4)^j*Binomial[k, j]*Binomial[k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)] ,{j, 0, n}]]; Flatten[Table[T[n, k],{n, 0, 8}, {k, 0, n}]] (* Detlef Meya, May 28 2024 *)

Formula

T(n, k) = if k = 0 then 1, otherwise 4^k*Sum_{j=0..n} (5/4)^j * binomial(k, j) * binomial(k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)). - Detlef Meya, May 28 2024

A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).

Original entry on oeis.org

1, 3, 33, 387, 4737, 59523, 761121, 9854211, 128772609, 1694927619, 22437369633, 298419470979, 3984500221569, 53376363001731, 717044895641121, 9656091923587587, 130310873022310401, 1761872309456567811, 23861153881099854369, 323634591584064809859
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Crossrefs

Cf. A299864, A245926, A084768, A245927, A253283.
Cf. A300945, A300946.

Programs

  • Maple
    seq(simplify( (-1)^n*hypergeom([-n, n], [1], 4)), n = 0..20); # Peter Bala, Apr 18 2024
  • Mathematica
    a[n_] := (-1)^n Hypergeometric2F1[-n, n, 1, 4]; Table[a[n], {n, 0, 19}]

Formula

From Vaclav Kotesovec, Jul 05 2018: (Start)
Recurrence: n*(2*n-3)*a(n) = 2*(14*n^2 - 28*n + 11)*a(n-1) - (n-2)*(2*n-1)*a(n-2).
a(n) ~ 2^(-3/2) * 3^(1/4) * (7 + 4*sqrt(3))^n / sqrt(Pi*n). (End)
From Peter Bala, Apr 18 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k-1)*3^k = R(n, 3) for n >= 1, where R(n, x) denotes the n-th row polynomial of A253283.
a(n) = 3*n* hypergeom([1 - n, n + 1], [2], -3) for n >= 1.
a(n) = (1/2)*(LegendreP(n, 7) - LegendreP(n-1, 7)) for n >= 1.
a(n) = [x^n] ( (1 - x)/(1 - 4*x) )^n.
It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
G.f.: (sqrt(x^2 - 14*x + 1) - x + 1)/(2*sqrt(x^2 - 14*x + 1)) = 1 + 3*x + 33*x^2 + 387*x^3 + .... (End)

A299845 a(n) = hypergeom([-n, n - 1], [1], -4).

Original entry on oeis.org

1, 1, 25, 425, 7025, 116625, 1951625, 32903225, 558265825, 9522632225, 163160773625, 2806202183625, 48420275891025, 837813745045425, 14531896733426025, 252593595973313625, 4398859688478578625, 76733590756134492225, 1340547988367851940825, 23451231922182584693225
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Links

Crossrefs

Cf. A299506, A243946, A084769, A243947.
Cf. A300945, A300946.

Programs

  • Maple
    f:= gfun:-rectoproc({4*n*(n-2)^2*a(n)+4*(n-1)^2*(n-3)*a(n-2)-4*(2*n-3)*(9*n^2-27*n+17)*a(n-1)=0,
a(0)=1,a(1)=1,a(2)=25},a(n),remember):
map(f, [$0..100]); # Robert Israel, Mar 21 2018
  • Mathematica
    a[n_] := Hypergeometric2F1[-n, n - 1, 1, -4]; Table[a[n], {n, 0, 19}]
a[0]:=1; a[1]:=1; a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, May 22 2024 *)

Formula

4*n*(n-2)^2*a(n) + 4*(n-1)^2*(n-3)*a(n-2) - 4*(2*n-3)*(9*n^2-27*n+17)*a(n-1) = 0. - Robert Israel, Mar 21 2018
a(n) ~ 2^(-3/2) * 5^(3/4) * phi^(6*n - 3) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 05 2018
a(n) = 4^n*Sum_{k=0..n} (5/4)^k*Gamma(n + 1)*Gamma(n - 1)/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1)) for n >= 2. - Detlef Meya, May 22 2024

A299864 a(n) = (-1)^n*hypergeom([-n, n - 1/2], [1], 4).

Original entry on oeis.org

1, 1, 19, 239, 3011, 38435, 496365, 6470385, 84975315, 1122708899, 14906800361, 198740733581, 2658870294349, 35677678567549, 479965685669059, 6471364940381007, 87425255326277907, 1183139999323074963, 16036589185819644633, 217668383345249016045
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Links

Crossrefs

Cf. A299507, A245926, A084768, A245927.
Cf. A300945, A300946.

Programs

  • Maple
    seq((-1)^n*orthopoly[P](n,0,-3/2,-7),n=0..100); # Robert Israel, Mar 21 2018
  • Mathematica
    a[n_] := (-1)^n Hypergeometric2F1[-n, n - 1/2, 1, 4]; Table[a[n], {n, 0, 19}]

Formula

From Robert Israel, Mar 21 2018: (Start)
a(n) = JacobiP(n,0,-3/2,-7).
n*(2*n-3)*(4*n-7)*a(n)+(2*n-5)*(n-1)*(4*n-3)*a(n-2)-(4*n-5)*(28*n^2-70*n+39)*a(n-1) = 0. (End)
a(n) ~ sqrt(3) * (1 + sqrt(3))^(4*n - 1) / (2^(2*n + 1) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 05 2018

A299506 a(n) = hypergeom([-n, n - 1/2], [1], -4).

Original entry on oeis.org

1, 3, 43, 661, 10515, 171097, 2828101, 47284251, 797456947, 13540982665, 231188344401, 3964874384863, 68252711769373, 1178662654873191, 20409993947488075, 354260920943874245, 6161735337225790035, 107368528677807960185, 1873946997372948997345, 32754419073618998202975
Offset: 0

Views

Author

Peter Luschny, Mar 16 2018

Keywords

Crossrefs

Cf. A299845, A243946, A084769, A243947.
Cf. A300945, A300946.

Programs

  • Mathematica
    a[n_] := Hypergeometric2F1[-n, n - 1/2, 1, -4]; Table[a[n], {n, 0, 19}]
a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1/2])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1/2]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, May 22 2024 *)

Formula

From Vaclav Kotesovec, Jul 05 2018: (Start)
Recurrence: n*(2*n - 3)*(4*n - 7)*a(n) = 9*(4*n - 5)*(4*n^2 - 10*n + 5)*a(n-1) - (n-1)*(2*n - 5)*(4*n - 3)*a(n-2).
a(n) ~ 2^(-3/2) * sqrt(5) * phi^(6*n - 3/2) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)
a(n) = 4^n*Sum_{k=0..n} (5/4)^k*(Gamma(n + 1)*Gamma(n - 1/2))/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1/2)). - Detlef Meya, May 22 2024
Showing 1-5 of 5 results.

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