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 11-14 of 14 results.

A065922 Bessel polynomial {y_n}'(4).

Original entry on oeis.org

0, 1, 27, 846, 32290, 1472535, 78444261, 4789283212, 329976556596, 25336918039005, 2145912573891295, 198763621138900026, 19988975122377164982, 2169175884299414423251, 252661578519463668740745, 31442098485128401965118680, 4163361054820272025075769896
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

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

Links

Crossrefs

Cf. A001514, A065920, A065921.

Programs

  • Mathematica
    Table[Sum[(n+k+1)!*2^(k-1)/((n-k-1)!*k!), {k,0,n-1}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==((2n-1)^3 a[n-1]+n^2 a[n-2])/ (n-1)^2}, a,{n,20}] (* Harvey P. Dale, May 23 2016 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!*2^(k-1)/((n-k-1)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

Recurrence: (n-1)^2*a(n) = (2*n - 1)^3*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(3*n-3/2) * n^(n+1) / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(1/2)_{n}*8^(n - 1)* hypergeometric1f1(1 - n, -2*n, 1/2).
E.g.f.: ((1 - 8*x)^(3/2) + 4*x*(1 - 8*x)^(1/2) + 24*x - 1)* exp((1 - sqrt(1 - 8*x))/4)/(16*(1 - 8*x)^(3/2)). (End)
G.f.: (t/(1-t)^3)*hypergeometric2f0(2,3/2; - ; 8*t/(1-t)^2). - G. C. Greubel, Aug 16 2017

A006199 Bessel polynomial {y_n}'(-1).

Original entry on oeis.org

0, 1, -3, 21, -185, 2010, -25914, 386407, -6539679, 123823305, -2593076255, 59505341676, -1484818160748, 40025880386401, -1159156815431055, 35891098374564105, -1183172853341759129, 41372997479943753582, -1529550505546305534414, 59608871544962952539335
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Absolute values give partitions into pairs.

References

  • G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000806, A001514, A065707, A065920, A065921, A065922.

Programs

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

Formula

a(n) = A000806(n) + (n-1) * A000806(n-1). - Sean A. Irvine, Jan 23 2017
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(1/2){n} * (-2)^(n-1) * hyergeometric1f1(1-n; -2*n; -2), where (a){n} is the Pochhammer symbol.
E.g.f.: (1+2*x)^(-3/2)*( (1+2*x)^(3/2) - x*(1+2*x)^(1/2) - x -1) * exp(sqrt(1+2*x) - 1), for offset 0. (End)
G.f.: (x/(1-x)^3)*hypergeometric2f0(2,3/2; - ; -2*x/(1-x)^2), for offset 0. - G. C. Greubel, Aug 16 2017

A065707 Bessel polynomial {y_n}'(-2).

Original entry on oeis.org

0, 1, -9, 126, -2270, 49995, -1301139, 39066076, -1329148764, 50536328085, -2123542798685, 97722882268506, -4887863677728954, 264025383760041631, -15317578742680490535, 949914821498248213560, -62707584375936061905464, 4390358319593012839913001
Offset: 0

Views

Author

N. J. A. Sloane, Dec 08 2001

Keywords

References

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

Links

Crossrefs

Cf. A001514, A065920, A065921, A065922, A065707, A006199.

Programs

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

Formula

From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(1/2)_{n}*(-4)^(n - 1)* hypergeometric1f1(1 - n, -2*n, -1).
E.g.f.: ((1 + 4*x)^(3/2) - 2*x*(1 + 4*x)^(1/2) - 1)* exp((sqrt(1 + 4*x) -1)/2)/(4*(1 + 4*x)^(3/2)). (End)
G.f.: (x/(1-x)^3)*hypergeometric2f0(2,3/2; - ; -4*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A144513 a(n) = Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k).

Original entry on oeis.org

2, 18, 162, 1670, 19980, 274932, 4296278, 75324762, 1466031690, 31386435410, 733391707752, 18578222154648, 507246285802802, 14851746921266010, 464244744007818090, 15431886798641124662, 543593886328031841828, 20228083875146926867932, 792934721766833544369830
Offset: 0

Views

Author

N. J. A. Sloane, Dec 16 2008

Keywords

Links

Crossrefs

Equals 2*A001514 (with a different offset).
Cf. A001515, A144498, A144514.

Programs

  • Maple
    f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..50)];
  • PARI
    {a(n) = sum(k=0, n, (n+k+2)!/((n-k)!*k!*2^k))} \\ Seiichi Manyama, Apr 07 2019

Formula

n^2*a(n) = (2*n+1)*(n^2+n+1)*a(n-1) + (n+1)^2*a(n-2). - Seiichi Manyama, Apr 07 2019
Previous Showing 11-14 of 14 results.

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

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