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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.

User: Yifan Zhang

Yifan Zhang's wiki page.

Yifan Zhang has authored 2 sequences.

A349466 Expansion of 1/((1-12*x)*(1-16*x)*(1-18*x)*(1-24*x)).

Original entry on oeis.org

1, 70, 3100, 111160, 3529456, 103663840, 2887307200, 77450369920, 2021488750336, 51703366274560, 1302470537804800, 32436048076257280, 800745898476630016, 19636648385968660480, 479101382689537638400, 11643791435175823114240, 282140675279022464106496
Offset: 0

Views

Author

Hsin-Hui Judy Chiang, Yifan Zhang and Wei Wang, Nov 18 2021

Keywords

Comments

a(n) = p(n+4,4)*(4!)^(n+1) where p(n+4,4) represents the probability that, given n+4 random numbers in [0, 1], there exists a 4-tuple whose sum is smaller than 1. A recurrence formula for p(n,k) is p(n, k) = (1/k)*p(n-1, k-1) + (1-1/k)*p(n-1, k). The generating function for p(n,k) is Sum_{n=k..oo} p(n,k)x^n = (x^k)/(k!*(1-x)*(1-(1/2)*x)*(1-((k-1)/k)*x)). The explicit formula for p(n,k) is p(n,k)= 1+(1/(k-1)!)*Sum_{i=1..(k-1)} ((-1)^(k-i))*binomial(k-1, i) * (i^n)* ((i+1)^(-n+k-1)).

Links

Crossrefs

Other sequences for p(n+k,k)*(k!)^(n+1) include: A000225 (k=2), A016765 (k=3).
Cf. A000079, A028212.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - 12 x) (1 - 16 x) (1 - 18 x) (1 - 24 x)), {x, 0, 20}],  x]
LinearRecurrence[{70,-1800,20160,-82944},{1,70,3100,111160},20] (* Harvey P. Dale, Apr 30 2023 *)
  • Python
    def A349466(n): return 24*24**n + 64*2**(4*n) - 81*18**n - 6*12**n # Chai Wah Wu, Dec 27 2021

Formula

a(n) = 24^(n+1) - (2^n)*(3^(2*n+4)) - (2^(2*n+1))*(3^(n+1)) + 2^(4*n+6).
G.f.: 1/((1-(1/2)*4!*x)*(1-(2/3)*4!*x)*(1-(3/4)*4!*x)*(1-4!*x)).
a(n) = 2^n * A028212(n) = A000079(n) * A028212(n).
E.g.f.: exp(12*x)*(24*exp(12*x) - 81*exp(6*x) + 64*exp(4*x) - 6). - Stefano Spezia, Nov 21 2021
a(n) = 24*24^n + 64*2^(4*n) - 81*18^n - 6*12^n. - Chai Wah Wu, Dec 27 2021

A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.

Original entry on oeis.org

1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483
Offset: 0

Views

Author

Yifan Zhang, Dec 13 2020

Keywords

References

  • Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.

Links

Crossrefs

Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).
Cf. A026000, A114496.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*Binomial[2n+k,k]*2^k,{k,0,n}],{n,0,20}] (* or *)
Table[Hypergeometric2F1[-n,1+2 n,1,-2],{n,0,20}] (* Stefano Spezia, Dec 17 2020 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021

Formula

a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
From Vaclav Kotesovec, May 11 2021: (Start)
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)

Extensions

More terms from Stefano Spezia, Dec 17 2020

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