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 11-17 of 17 results.

A065947 Bessel polynomial {y_n}''(3).

Original entry on oeis.org

0, 0, 6, 300, 13320, 620130, 31406550, 1743174216, 105889417200, 7010411889690, 503353562247360, 39003404559533700, 3246506259033473436, 289042023964190515200, 27418894569798460848210, 2761554229456140638184840, 294364593823858690215256200
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Crossrefs

Cf. A001516, A001518.

Programs

  • Mathematica
    Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*6^(n - 2)* Hypergeometric1F1[2 - n, -2*n, 2/3], {n, 2, 50}]] (* G. C. Greubel, Aug 14 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(3/2)^k), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2){n}*6^(n - 2)*hypergeometric1F1(2-n, -2*n, 2/3), where (a){n} is the Pochhammer symbol.
E.g.f.: (-1/81)*(1 - 6*x)^(-5/2)*((171*x^2 - 90*x + 8)*sqrt(1 - 6*x) + (54*x^3 - 648*x^2 + 114*x - 8))*exp((1 - sqrt(1 - 6*x))/3). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2F0(3,5/2; - ; 6*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065948 Bessel polynomial {y_n}''(-3).

Original entry on oeis.org

0, 0, 6, -240, 9540, -415590, 20134590, -1082674404, 64221641820, -4173853100670, 295282282905720, -22605059036265420, 1862664627479732076, -164425432052147568120, 15483794266369962976170, -1549617160894627918342620, 164264715996348003982855020
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Crossrefs

Cf. A001518, A001516.

Programs

  • Mathematica
    Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*(-6)^(n - 2)* Hypergeometric1F1[2 - n, -2*n, -2/3], {n, 2, 50}]] (* G. C. Greubel, Aug 15 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(-3/2)^k ), ", ")) \\ G. C. Greubel, Aug 15 2017

Formula

From G. C. Greubel, Aug 15 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2){n}*(-6)^(n - 2)* hypergeometric1f1(2 - n; -2*n; -2/3), where (a){n} is the Pochhammer symbol.
E.g.f.: (-1/81)*(1 + 6*x)^(-5/2)*((-99*x^2 - 54*x - 4)*sqrt(1 + 6*x) + (-54*x^3 + 66*x + 4))*exp(-(1 - sqrt(1 + 6*x))/3). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; -6*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065949 Bessel polynomial {y_n}'''(0).

Original entry on oeis.org

0, 0, 0, 90, 630, 2520, 7560, 18900, 41580, 83160, 154440, 270270, 450450, 720720, 1113840, 1670760, 2441880, 3488400, 4883760, 6715170, 9085230, 12113640, 15939000, 20720700, 26640900, 33906600, 42751800, 53439750, 66265290, 81557280, 99681120, 121041360, 146084400
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Crossrefs

Cf. A001518, A001516.

Programs

  • Mathematica
    Drop[90*Binomial[Range[40]-3,6],5] (* Harvey P. Dale, Sep 20 2013 *)
  • PARI
    for(n=0,50, print1(90*binomial(n+3,6), ", ")) \\ G. C. Greubel, Aug 15 2017

Formula

a(n) = 90 * C(n-3, 6) = 90 * A000579(n-3). - Ralf Stephan, Sep 03 2003
From Colin Barker, Aug 01 2013: (Start)
a(n) = ((-2+n)*(-1+n)*n*(1+n)*(2+n)*(3+n))/8.
G.f.: -90*x^3 / (x-1)^7. (End)
E.g.f.: (1/8)*x^3*(120 + 90*x + 18*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 15 2017

Extensions

More terms from Colin Barker, Aug 01 2013

A065950 Bessel polynomial {y_n}'''(1).

Original entry on oeis.org

0, 0, 0, 90, 3150, 81900, 1992060, 48771450, 1237774230, 32978969100, 927339227100, 27566149731120, 866148362679600, 28735959507074820, 1005105838958594100, 36999204981675832350, 1430792213377354462530, 58019598569681129648700
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Crossrefs

Cf. A001516, A001518.

Programs

  • Magma
    [0,0,0] cat [(&+[Binomial(n-3,k)*Factorial(n+k+3)/(2^(k+3) * Factorial(n-3)): k in [0..n-3]]): n in [3..30]]; // G. C. Greubel, Sep 23 2023
  • Mathematica
    Join[{0,0,0}, Table[6*Binomial[n,3]*Pochhammer[1/2,n]*2^n* Hypergeometric1F1[3-n,-2*n,2], {n,3,50}]] (* G. C. Greubel, Aug 15 2017 *)
CoefficientList[Series[(90*t^3/(1-t)^7)*HypergeometricPFQ[{4, 7/2}, {}, 2*t/(1-t)^2], {t,0,50}], t] (* G. C. Greubel, Aug 16 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n-3, ((n+k+3)!/(2^(k+3)*k!*(n-k-3)!))), ", ")) \\ G. C. Greubel, Aug 15 2017
  • SageMath
    def A065950(n): return sum(binomial(n-3,k)*rising_factorial(n-2,k+6)//2^(k+3) for k in range(n-2))
[A065950(n) for n in range(31)] # G. C. Greubel, Sep 23 2023

Formula

a(n) = 6*binomial(n, 3)*(1/2){n}*2^n*hypergeometric1f1(3-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 15 2017
G.f.: (90*x^3/(1-x)^7)*hypergeometric2f0(4,7/2; - ; 2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) ~ 2^(n + 1/2) * n^(n+3) / exp(n-1). - Vaclav Kotesovec, Jun 09 2019

A065951 Bessel polynomial {y_n}'''(-1).

Original entry on oeis.org

0, 0, 0, 90, -1890, 36540, -729540, 15507450, -353908170, 8680615020, -228436914420, 6431738433120, -193144902350400, 6166945337372820, -208728050864680620, 7467661073819689470, -281666767117960443870, 11173071188540083124700
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Crossrefs

Cf. A001518, A001516.

Programs

  • Mathematica
    Join[{0, 0, 0}, Table[48*Binomial[n, 3]*Pochhammer[1/2, n]*(-2)^(n - 3)* Hypergeometric1F1[3 - n, -2*n, -2], {n,3,50}]] (* G. C. Greubel, Aug 15 2017 *)
CoefficientList[Series[(90*t^3/(1 - t)^7)*HypergeometricPFQ[{4, 7/2}, {}, -2*t/(1 - t)^2], {t, 0, 50}], t] (* G. C. Greubel, Aug 16 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n-3, ((n+k+3)!/(8*(n-k-3)!*k!))*(-2)^k ), ", ")) \\ G. C. Greubel, Aug 15 2017

Formula

a(n) = 48*binomial(n,3)*(1/2){n}*(-2)^(n - 3)*hypergeometric1f1(3-n, -2*n, -2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 15 2017
G.f.: (90*x^3/(1-x)^7)*hypergeometric2f0(4,7/2; - ; -2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
D-finite with recurrence (-n+3)*a(n) +(-2*n^2+11)*a(n-1) +(-2*n^2+8*n+3)*a(n-2) +(n+1)*a(n-3)=0. - R. J. Mathar, Jul 25 2022

A078976 Numerator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 2, 37, 261, 298, 559, 5888, 318511, 5102064, 5420575, 10522639, 205350716, 18492087079, 462507527691, 480999614770, 943507142461, 26899199603678, 3390242657205889, 115295149544603904, 118685392201809793, 233980541746413697, 8775965436819116582, 1421940381306443299981
Offset: 1

Views

Author

Benoit Cloitre, Dec 19 2002

Keywords

Crossrefs

Cf. A069951, A001518, A007676, A078977 (denominators).

Programs

  • Mathematica
    Convergents[Exp[2/3], 25] // Numerator (* Amiram Eldar, May 09 2025 *)
  • PARI
    default(realprecision,100); /* large enough */
a(n)=contfracpnqn(contfrac(exp(2/3), n))[1,1]
vector(30,n,a(n))

Formula

Special cases : a(5k+1) = A001518(3k); a(5k+3) = A001518(3k+2).

A078977 Denominator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 1, 19, 134, 153, 287, 3023, 163529, 2619487, 2783016, 5402503, 105430573, 9494154073, 237459282398, 246953436471, 484412718869, 13810509564803, 1740608617884047, 59194503517622401, 60935112135506448
Offset: 1

Views

Author

Benoit Cloitre, Dec 19 2002

Keywords

Crossrefs

Cf. A069951, A065923, A001518, A007677, A078976.

Programs

  • Mathematica
    Denominator[Convergents[E^(2/3),20]] (* Harvey P. Dale, Dec 01 2013 *)
  • PARI
    a(n)=component(component(contfracpnqn(contfrac(exp(2/3),n)),1),2) \\ (Warning: this will give only a limited number of correct terms, depending on the precision used. - The Editors, Oct 13 2009. See A078976 for better code.)

Formula

Special cases : a(5k+1)=abs(A065923(3k)); a(5k+3)=abs(A065923(3k+2)) where A065923(n)=y(n, -3) where y(n, x)=sum (k=0, n, (n+k)!*(x/2)^k/((n-k)!*k!))
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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