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.

Previous Showing 11-18 of 18 results.

A178119 Expansion of 1/(1-2x-x^2/(1-4x-3x^2/(1-6x-5x^2/(1-8x-7x^2/(1-...))))) (continued fraction).

Original entry on oeis.org

1, 2, 5, 16, 64, 308, 1727, 11008, 78244, 611060, 5184338, 47366320, 462782080, 4807659368, 52853722811, 612426360832, 7453621425532, 94997205901940, 1264555335831662, 17540102647480336, 252979919852470672
Offset: 0

Views

Author

Paul Barry, May 20 2010

Keywords

Comments

Hankel transform is A057863. First column of A178121.

Examples

			G.f. = 1 + 2*x + 5*x^2 + 16*x^3 + 64*x^4 + 308*x^5 + 1727*x^6 + 11008*x^7 + ...

Formula

G.f.: 1/(Q(0)-x) where Q(k) = 1 - x*(2*k+1)/( 1 - x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013

A185998 Expansion of 1/(1-x-x^2/(1-2x-3x^2/(1-4x-5x^2/(1-6x-7x^2/(1-8x-9x^2/(1-...)))))) (continued fraction).

Original entry on oeis.org

1, 1, 2, 5, 16, 64, 308, 1727, 11008, 78244, 611060, 5184338, 47366320, 462782080, 4807659368, 52853722811, 612426360832, 7453621425532, 94997205901940, 1264555335831662, 17540102647480336, 252979919852470672, 3786896497867089656, 58733519954606892278, 942372338052077455168
Offset: 0

Views

Author

Paul Barry, Feb 08 2011

Keywords

Comments

First column of A185997. Moment sequence for orthogonal polynomials of A185996. Hankel transform is A057863.

Crossrefs

Cf. A178119.

Formula

G.f.: 1/(1-x/(1-x/(1-x/(1-3x/(1-x/(1-5x/(1-x/(1-7x/(1-x/(1-9x/(1-...))))))))))) (continued fraction).

A089626 a(n) = 1/h(n) where {h(n)} is the Hankel transform of {t(n)}; t(n) is defined by the expansion of tan(x)= Sum_n>0, t(n)*x^(2*n-1); |x|

Original entry on oeis.org

1, 45, 4465125, 6272287562165625, 438120013555654794702228515625, 3943988517696329309474874414036059896739501953125, 9860368980530253649041813027973243717504383071655695011832599639892578125
Offset: 1

Views

Author

Philippe Deléham, Dec 31 2003

Keywords

Comments

t(n)= (2^(2*n)-1)*2^(2*n)*B_n /(2*n)! B_n: numbers of Bernoulli, sequence 1/6, 1/30, 1/42, 1/30, 5/66, ... example:n=2, a(2)= 1/det|1, 1/3|1/3, 2/15|= 1/(1/45)=45 See A001906 for the definition of Hankel transform.

Crossrefs

Cf. A057863.

Formula

a(n) = (4*n-3)^1*(4*n-5)^2*...*3^(2*n-2)*1^(2*n-1).
a(n) = A057863(2*n-1). - Vaclav Kotesovec, Oct 16 2015

A137565 Hankel transform of quadruple factorial numbers A001813.

Original entry on oeis.org

1, 8, 3072, 141557760, 1461147874099200, 5429466527263925207040000, 10652562037579218110269404217344000000, 15215364593060190936848463156185060517032755200000000
Offset: 0

Views

Author

Paul Barry, Jan 26 2008, Apr 25 2008

Keywords

Formula

a(n)=Product{k=0..n-1, (8(k+1)(2k+1))^(n-k)}.
a(n)=8^C(n+1,2)*Product{k=1..n, k!*Product{i=1..k-1, 2i+1}}; a(n)=8^C(n+1,2)*A000178(n)*A057863(n).

Extensions

Edited by N. J. A. Sloane, May 14 2008 at the suggestion of R. J. Mathar.

A178123 Expansion of 1/(1-x-x^2/(1-2x-3x^2/(1-3x-5x^2/(1-4x-7x^2/(1-... (continued fraction).

Original entry on oeis.org

1, 1, 2, 5, 16, 61, 269, 1337, 7354, 44155, 286397, 1990427, 14725738, 115356349, 952592288, 8261093885, 74994333994, 710656444489, 7012302313061, 71892455879393, 764331907463476, 8411953721081635, 95684448908132498
Offset: 0

Views

Author

Paul Barry, May 20 2010

Keywords

Comments

First column of A178125. Hankel transform is A057863.

A263430 a(n) = Product_{k=0..n} (4*k+1)^(n-k).

Original entry on oeis.org

1, 1, 5, 225, 131625, 1309010625, 273380323978125, 1427352844030287890625, 216119240915841469025244140625, 1079864992142473709995957417730712890625, 199639840782299404795675492100337942688751220703125
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 18 2015

Keywords

Links

Crossrefs

Cf. A000178, A057863, A263416.

Programs

  • Magma
    [(&*[(4*k+1)^(n-k): k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 25 2018
  • Mathematica
    Table[Product[(4*k+1)^(n-k),{k,0,n}],{n,0,10}]
  • PARI
    for(n=0,10, print1(prod(k=0,n, (4*k+1)^(n-k)), ", ")) \\ G. C. Greubel, Aug 25 2018

Formula

a(n) ~ A^(1/8) * 2^(n^2 + 3*n/2 + 1/8) * Pi^(n/2 + 1/8) * n^(n^2/2 + n/4 - 5/96) / (Gamma(1/4)^(n + 1/4) * exp(3*n^2/4 + n/4 + 1/96 - C/(4*Pi))), where A = A074962 is the Glaisher-Kinkelin constant and C = A006752 = is Catalan's constant.

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4*n-1,4*n-2,...,2,1) and mu is the square n^n.

Original entry on oeis.org

1, 16, 4362327818240, 19265181532031090042534736325278852710400, 830325323503973129435791248069702287019820905338483131168940909920954227594481411031040
Offset: 0

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Author

Alejandro H. Morales, Sep 05 2017

Keywords

Comments

The number of standard Young tableaux of a fixed skew shape has a determinantal formula, the Jacobi-Trudi formula. It is rare when a family of skew shapes has a product formula for the number of standard Young tableaux. This product formula has independently been proved using P-Schur functions (by DeWitt) and using the Naruse hook-length formula for skew shapes (by Morales, Pak and Panova).

Examples

			a(1)=16 since there are 16 standard Young tableaux of skew shape 321/1 since this is the same as the number of standard Young tableaux of straight shape 321 given by the hook-length formula: 16 = 6!/(3^2*5).

Links

Crossrefs

Cf. A000178, A057863, A008793, A291871, A000085, A061343, A005118.

Programs

  • Maple
    b:=n->mul(factorial(i),i=1..n-1):
c:=n->mul(doublefactorial(2*i-1),i=1..n-1):
a:=n->factorial(binomial(4*n,2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)):
seq(a(n),n=0..9);
  • Sage
    def b(n): return mul([factorial(i) for i in range(1,n)])
def d(n): return factorial(n+1)/(2^((n+1)/2)*factorial((n+1)/2))
def c(n): return mul([d(2*i-1) for i in range(1,n)])
def a(n):
    return factorial(binomial(4*n,2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n))
[a(n) for n in range(10)]

Formula

a(n) = (binomial(4*n,2)-n^2)!*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)) where b(n) = 1!*2!*...*(n-1)! is the superfactorial A000178(n-1), and c(n) = 1!!*3!!*...*(2*n-3)!! is super doublefactorial A057863(n-1).
a(n) ~ sqrt(Pi) * 3^(9*n^2 - 3*n/2 - 1/24) * 7^(7*n^2 - 2*n + 1/2) * exp(7*n^2/2 - 2*n + 23/56) * n^(7*n^2 - 2*n + 7/8) / (A^(3/2) * 2^(33*n^2 - 6*n - 7/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

A372509 Numbers that can be written as a product of double factorials of odd numbers.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 81, 105, 135, 225, 243, 315, 405, 675, 729, 945, 1215, 1575, 2025, 2187, 2835, 3375, 3645, 4725, 6075, 6561, 8505, 10125, 10395, 10935, 11025, 14175, 18225, 19683, 23625, 25515, 30375, 31185, 32805, 33075, 42525, 50625
Offset: 1

Views

Author

Ilya Gutkovskiy, May 04 2024

Keywords

Crossrefs

Cf. A001013, A001147, A025487, A057863.

Programs

  • Mathematica
    kmax = 60000;
maxExponent = 10;
ff = Select[TakeWhile[Range[3, kmax], #!! < kmax &]!!, OddQ];
lg = Length@ff;
iter = Sequence @@ Table[{x[i], 0, maxExponent}, {i, 1, lg}];
seq = Table[k = Times @@ Table[ff[[i]]^x[i] , {i, 1, lg}]; If[k <= kmax, k, Nothing], Evaluate@iter] // Flatten // Union (* Jean-François Alcover, May 11 2024 *)
Previous Showing 11-18 of 18 results.

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