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.

Showing 1-4 of 4 results.

A001820 Central factorial numbers: 2nd subdiagonal of A008955.

Original entry on oeis.org

1, 14, 273, 7645, 296296, 15291640, 1017067024, 84865562640, 8689315795776, 1071814846360896, 156823829909121024, 26862299458337581056, 5325923338791614078976, 1210310405427816646041600, 312542036038910895995289600, 91018216923341770801874534400
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n-2) is the coefficient of x^3 in Product_{k=0..n} (x + k^2).

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A049033.
Third right-hand column of triangle A008955.

Programs

  • Maple
    seq(2*Stirling1(n+3, 1)*Stirling1(n+3, 5)-2*Stirling1(n+3, 2)*Stirling1(n+3, 4)+Stirling1(n+3, 3)^2, n=0..20); # Mircea Merca, Apr 03 2012
  • Mathematica
    Table[StirlingS1[n+3, 3]^2 - 2*StirlingS1[n+3, 2]*StirlingS1[n+3, 4] + 2*StirlingS1[n+3, 1]*StirlingS1[n+3, 5], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

Formula

a(n) = s(n+3,3)^2 - 2*s(n+3,2)*s(n+3,4) + 2*s(n+3,1)*s(n+3,5), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012
a(n) = (3*n^2 + 6*n + 5)*a(n-1) - (n^2 + n + 1)*(3*n^2 + 3*n + 1)*a(n-2) + n^6*a(n-3). - Vaclav Kotesovec, Feb 23 2015
a(n) ~ Pi^5 * n^(2*n+5) / (60 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

A001824 Central factorial numbers: 1st subdiagonal of A008956.

Original entry on oeis.org

1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125, 55399444912646408707007883750, 46636497509226736668824289999375
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			(arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...

References

  • T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002455, A001825, A049033.
Right-hand column 2 in triangle A008956.

Programs

  • Mathematica
    a[n_] = (2n+1)!!^2 (Pi^2 - 2 PolyGamma[1, n+3/2])/8; a /@ Range[0, 12] // Simplify (* Jean-François Alcover, Apr 22 2011, after Joe Keane *)
With[{nn=30},Take[(CoefficientList[Series[ArcSin[x]^3,{x,0,nn}], x] Range[0,nn-1]!)/6,{4,-1,2}]] (* Harvey P. Dale, Feb 05 2012 *)

Formula

E.g.f.: (arcsin x)^3; that is, a_k is the coefficient of x^(2*k+3) in (arcsin x)^3 multiplied by (2*k+3)! and divided by 6. - Joe Keane (jgk(AT)jgk.org)
a(n) = ((2*n+1)!!)^2 * Sum_{k=0..n} (2*k+1)^(-2).
a(n) ~ Pi^2*n^2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
(-1)^(n-1)*a(n-1) is the coefficient of x^2 in Product_{k=1..2*n} (x + 2*k - 2*n - 1). - Benoit Cloitre and Michael Somos, Nov 22 2002
a(n) = det(V(i+2,j+1), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices (A008958). - Mircea Merca, Apr 06 2013
Recurrence: a(n) = 2*(4*n^2+1)*a(n-1) - (2*n-1)^4*a(n-2). - Vladimir Reshetnikov, Oct 13 2016
Limit_{n->infinity} a(n)/((2n+1)!!)^2 = Pi^2/8. - Daniel Suteu, Oct 31 2017

Extensions

More terms from Joe Keane (jgk(AT)jgk.org)

A002455 Central factorial numbers: unsigned 1st subdiagonal of A182867.

Original entry on oeis.org

0, 1, 20, 784, 52480, 5395456, 791691264, 157294854144, 40683662475264, 13288048674471936, 5349739088314368000, 2603081566154391552000, 1506057980251484454912000, 1021944601582419125993472000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			(arcsin x)^4 = x^4 + 2/3*x^6 + 7/15*x^8 + 328/945*x^10 + ...

References

  • B. Berndt, Ramanujan's Notebooks, Part I, page 263.
  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001819, A001824, A001825, A049033.

Programs

  • GAP
    List([0..20], n-> 4^(n-1)*(Factorial(n))^2*Sum([1..n], k-> 1/k^2)); # G. C. Greubel, Jul 04 2019
  • Magma
    [0] cat [4^(n-1)*(Factorial(n))^2*(&+[1/k^2: k in [1..n]]): n in [1..20]]; // G. C. Greubel, Jul 04 2019
  • Maple
    A002455 := proc(n)
    arcsin(x)^4;
    coeftayl(%,x=0,2*n+2)*(2*n+2)!/4! ;
end proc:
seq(A002455(n),n=0..20) ; # R. J. Mathar, Jan 20 2025
  • Mathematica
    nmax = 13; coes = CoefficientList[ Series[ ArcSin[x]^4, {x, 0, 2*nmax + 2}], x]* Range[0, 2*nmax + 2]!/24; a[n_] := coes[[2*n + 3]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 08 2011 *)
Table[4^(n-1)*(n!)^2*HarmonicNumber[n,2], {n,0,20}] (* G. C. Greubel, Jul 04 2019 *)
  • PARI
    a(n)=if(n<0,0,(2*n+2)!*polcoeff(asin(x+O(x^(2*n+3)))^4/4!,2*n+2))
  • PARI
    a(n)=-(-1)^n*polcoeff(prod(k=0,2*n,x+2*k-2*n),3)
  • Sage
    [4^(n-1)*(factorial(n))^2*sum(1/k^2 for k in (1..n)) for n in (0..20)] # G. C. Greubel, Jul 04 2019

Formula

(-1)^(n-1)*a(n) is the coefficient of x^3 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: (arcsin x)^4; that is, a_k is the coefficient of x^(2*k+2) in (arcsin x)^4 multiplied by (2*k+2)! and divided by 4! Also a(n) = 2^(2*n-2)*(n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)
a(n) = 4*(2*n^2 - 2*n + 1)*a(n-1) - 16*(n-1)^4*a(n-2). - Vaclav Kotesovec, Feb 23 2015
a(n) ~ Pi^3 * 2^(2*n-2) * n^(2*n+1) / (3 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Extensions

More terms from Joe Keane (jgk(AT)jgk.org)

A001825 Central factorial numbers: 2nd subdiagonal of A008956.

Original entry on oeis.org

1, 35, 1974, 172810, 21967231, 3841278805, 886165820604, 261042753755556, 95668443268795341, 42707926241367380631, 22821422608929422854674, 14384681946935352617964750, 10562341153570752891930640875
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			(arcsin x)^5 = x^5 + 5/6*x^7 + 47/72*x^9 + 1571/3024*x^11 + ...

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001824, A002455, A049033.
Right-hand column 3 in triangle A008956.

Programs

  • Mathematica
    Table[(2*n+5)!/5! * SeriesCoefficient[ArcSin[x]^5,{x,0,2*n+5}], {n,0,20}] (* Vaclav Kotesovec, Feb 23 2015 *)

Formula

E.g.f.: (arcsin x)^5; that is, a_k is the coefficient of x^(2*k+5) in (arcsin x)^5 multiplied by (2*k+5)! and divided by 5!. - Joe Keane (jgk(AT)jgk.org)
(-1)^(n-2)*a(n-2) is the coefficient of x^4 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
a(n) = det(V(i+3,j+2), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices (A008958). - Mircea Merca, Apr 06 2013
a(n) = (12*n^2 + 12*n + 11)*a(n-1) - (4*n^2 + 3)*(12*n^2 + 1)*a(n-2) + (2*n - 1)^6*a(n-3). - Vaclav Kotesovec, Feb 23 2015
a(n) ~ Pi^4 * n^(2*n+4) * 2^(2*n-2) / (3*exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Extensions

More terms from Joe Keane (jgk(AT)jgk.org)
Showing 1-4 of 4 results.

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

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