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 11-16 of 16 results.

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

A117852 Mirror image of A098473 formatted as a triangular array.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 18, 6, 1, 70, 80, 36, 8, 1, 252, 350, 200, 60, 10, 1, 924, 1512, 1050, 400, 90, 12, 1, 3432, 6468, 5292, 2450, 700, 126, 14, 1, 12870, 27456, 25872, 14112, 4900, 1120, 168, 16, 1, 48620, 115830, 123552, 77616, 31752, 8820, 1680, 216, 18, 1
Offset: 0

Views

Author

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

Keywords

Examples

			Triangle begins:
    1;
    2,   1;
    6,   4,   1;
   20,  18,   6,   1;
   70,  80,  36,   8,   1;
  252, 350, 200,  60,  10,   1;
  ...

Links

Crossrefs

Cf. A098473.

Programs

  • Maple
    c:=n->binomial(2*n, n): T:=proc(n, k) if k<=n then binomial(n, k)*c(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; #
  • Mathematica
    Table[ Binomial[n, k]*Binomial[2*n - 2*k, n - k], {n,0,10}, {k,0,n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

Formula

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
T(n,k) = binomial(n,k)*A000984(n-k). - Philippe Deléham, Dec 12 2009
O.g.f.: 1/sqrt( (1 - x*t)*(1 - (x + 4)*t) ) = 1 + (2 + x)*t + (6 + 4*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 12 2007

A098411 Expansion of 1/(sqrt(1-4x)sqrt(1-12x)).

Original entry on oeis.org

1, 8, 72, 704, 7264, 77568, 847104, 9394176, 105334272, 1190899712, 13551235072, 154997784576, 1780378353664, 20522842062848, 237284128063488, 2750571189633024, 31956067676454912, 371997834879172608, 4337957919010062336, 50664706036388069376, 592558533060795039744
Offset: 0

Views

Author

Paul Barry, Sep 07 2004

Keywords

Comments

Nguyen and Taggart (see link) conjecture: det[a(i+j) for i,j=0..n] = b(n)*b(n+1)/2 with b(n) = A139685(n). - Peter Luschny, May 19 2015

Links

Crossrefs

Cf. A098410, A139685.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(Sqrt[1-4*x]*Sqrt[1-12*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
  • PARI
    x='x+O('x^66); Vec(1/sqrt(1-16*x+48*x^2)) \\ Joerg Arndt, May 11 2013
  • Sage
    a = lambda n: 4^n*hypergeometric([-n, 1/2], [1], -2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

Formula

G.f.: 1/sqrt(1-16x+48x^2).
E.g.f.: exp(8x)*BesselI(0, 4x).
a(n) = Sum_{k=0..n} 3^k*binomial(2k, k)*binomial(2(n-k), n-k).
D-finite with recurrence: n*a(n) +8*(1-2*n)*a(n-1) +48*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
a(n) ~ sqrt(3)*12^n/sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 15 2012
a(n) = 4^n*hypergeometric([-n, 1/2], [1], -2). - Peter Luschny, May 19 2015

A293491 a(n) = n! * [x^n] exp((n+2)*x)*BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 10 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A000984.

Links

Crossrefs

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

Formula

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017

A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)*16*(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

Original entry on oeis.org

1, 6, 76, 1260, 24276, 515592, 11721072, 280020312, 6945369860, 177358000248, 4635276570288, 123449340098448, 3339525750984528, 91535631253610400, 2537277723600799680, 71015600640006437040, 2004523477053308685540, 57003431104378084982040
Offset: 1

Views

Author

Bradley Klee, Aug 30 2018

Keywords

Comments

Area interior to the central loop of u = 2*H = x^2 + y^2 - (1/2)*(x^4 + y^4) equals to Pi*S(u), when u in [0,1/2].

Examples

			Singular Value: S(1/2) = 1/sqrt(2).
N=4, h=1/sqrt(2) Quantization: S(u) = (n+1/2)*h/N.
  n  |                  u
==================================================
  0  |  0.08544689553344134756293807606337...
  1  |  0.23840989875904155311088418238272...
  2  |  0.36638282702449450473835851051425...
  3  |  0.46595506694324457665483887176081...

References

  • E. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press, 2018, page 204.

Links

Crossrefs

Cf. A318417, A010503, A247719, A000888.

Programs

  • GAP
    a:=[1,6];; for n in [3..20] do a[n]:=(1/(n*(n-1)^2))*(12*(n-1)*(2*n-3)^2*a[n-1]-(128*(n-2)*(2*n-5)*(2*n-3)*a[n-2])); od; a; # Muniru A Asiru, Sep 24 2018
  • Mathematica
    RecurrenceTable[{(n-1)^2*n*a[n] - 12*(n-1)*(2*n-3)^2*a[n-1] + 128*(n-2)*(2*n-5)*(2*n-3)*a[n-2] == 0, a[1] == 1, a[2] == 6}, a, {n, 1, 1000}]

Formula

(n-1)^2*n*a(n) - 12*(n-1)*(2*n-3)^2*a(n-1) + 128*(n-2)*(2*n-5)*(2*n-3)*a(n-2) == 0.
a(n) = A000108(n-1)*A098410(n-1).

A358114 a(n) = [x^n] (16*x*(32*x - 3) + 1)^(-1/2).

Original entry on oeis.org

1, 24, 608, 16128, 443904, 12570624, 363708416, 10694295552, 318301929472, 9562594738176, 289380790960128, 8807948507676672, 269349580129173504, 8268747111256817664, 254668380196759928832, 7865254221563736096768, 243493498808268962660352, 7553805204299934842486784
Offset: 0

Views

Author

Peter Luschny, Nov 12 2022

Keywords

Crossrefs

Cf. A098430.

Programs

  • Maple
    ogf := (16*x*(32*x - 3) + 1)^(-1/2): ser := series(ogf, x, 20):
seq(coeff(ser, x, n), n = 0..17);
  • Mathematica
    a[n_] := 16^n * HypergeometricPFQ[{1/2, -n}, {1}, -1]; Array[a, 18, 0] (* Amiram Eldar, Nov 12 2022 *)

Formula

a(n) = 16^n * hypergeom([1/2, -n], [1], -1).
D-finite with recurrence a(n) = (24*(2*n - 1)*a(n - 1) - 512*(n - 1)*a(n - 2)) / n for n >= 2.
a(n) ~ 2^(5*n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Nov 12 2022
a(n) = 4^n*A098410(n). - R. J. Mathar, Jan 25 2023
Previous Showing 11-16 of 16 results.

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