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 1-10 of 12 results. Next

A375231 Interleaving A006040 and A228229.

Original entry on oeis.org

1, 1, 3, 9, 19, 82, 229, 1313, 4581, 32826, 137431, 1181737, 5772103, 57905114, 323237769, 3705927297, 23273119369, 300180111058, 2094580743211, 30018011105801, 230403881753211, 3632179343801922, 30413312391423853, 523033825507476769, 4744476733062121069, 88392716510763573962
Offset: 0

Views

Author

Stefano Spezia, Aug 06 2024

Keywords

Comments

Conjecture: a(n) is the permanent of the n X n matrix whose generic element is given by M_{i,j} = 1 if i = j or i + j = 1 (mod 2), with i,j in [n].

Crossrefs

Cf. A006040, A078126 (determinant), A228229.

Programs

  • Mathematica
    A006040[n_]:=Sum[(n!/(n - k)!)^2, {k, 0, n}]; A228229[n_]:= n!*(n + 1)!*Sum[ 1/(k!*(k + 1)!),{k,0,n}]; a[n_]:=If[OddQ[n],A006040[(n+1)/2],A228229[n/2]]; Array[a,26,0]

Formula

a(n) = A006040((n+1)/2) for odd n.
a(n) = A228229(n/2) for even n.

A096789 Decimal expansion of BesselI(1,2).

Original entry on oeis.org

1, 5, 9, 0, 6, 3, 6, 8, 5, 4, 6, 3, 7, 3, 2, 9, 0, 6, 3, 3, 8, 2, 2, 5, 4, 4, 2, 4, 9, 9, 9, 6, 6, 6, 2, 4, 7, 9, 5, 4, 4, 7, 8, 1, 5, 9, 4, 9, 5, 5, 3, 6, 6, 4, 7, 1, 3, 2, 2, 8, 7, 9, 8, 4, 6, 0, 8, 5, 4, 5, 0, 3, 7, 5, 3, 5, 3, 6, 1, 1, 8, 5, 1, 1, 6, 1, 2, 2, 1, 4, 7, 5, 9, 4, 2, 2, 8, 9, 2, 5, 2, 3, 7, 7, 5
Offset: 1

Views

Author

Robert G. Wilson v, Jul 09 2004

Keywords

Examples

			1.59063685463732906338225...

Links

Crossrefs

Cf. A000108, A070910, A228229, A348607.

Programs

  • Maple
    evalf(BesselI(1,2)). # R. J. Mathar, Oct 16 2015
  • Mathematica
    RealDigits[BesselI[1, 2], 10, 110][[1]]
(* Or *) RealDigits[ Sum[ n/(n!n!), {n, 0, Infinity}], 10, 110][[1]]
  • PARI
    besseli(1,2) \\ Charles R Greathouse IV, Feb 19 2014

Formula

Equals Sum_{k >= 0} k/k!^2.
Continued fraction expansion: 1/(1 - 1/(3 - 2/(7 - 6/(13 - 12/(21 - ... - n*(n-1)/(n^2+n+1 - ...)))))). For a sketch of the proof see A228229. Cf. A070910. - Peter Bala, Aug 19 2013
From Amiram Eldar, Jul 09 2023: (Start)
Equals exp(-2) * Sum_{k>=1} A000108(k)/(k-1)!.
Equals exp(2) * Sum_{k>=1} (-1)^(k+1) * A000108(k)/(k-1)!. (End)

A099597 Array T(n,k) read by antidiagonals: expansion of exp(x+y)/(1-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 19, 5, 1, 1, 6, 33, 82, 33, 6, 1, 1, 7, 51, 229, 229, 51, 7, 1, 1, 8, 73, 496, 1313, 496, 73, 8, 1, 1, 9, 99, 919, 4581, 4581, 919, 99, 9, 1, 1, 10, 129, 1534, 11905, 32826, 11905, 1534, 129, 10, 1, 1, 11, 163, 2377, 25733, 137431, 137431, 25733, 2377, 163, 11, 1
Offset: 0

Views

Author

Ralf Stephan, Oct 28 2004

Keywords

Comments

Rows are polynomials in n whose coefficients are in A099599.
From Peter Bala, Aug 19 2013: (Start)
The k-th superdiagonal sequence of this square array occurs as the sequence of numerators in the convergents to a certain continued fraction representation of the constant BesselI(k,2), where BesselI(k,x) is a modified Bessel function of the first kind:
Let d_k(n) = T(n,n+k) = n! * (n+k)! * Sum_{i=0..n} 1/(i!*(i+k)!) denote the sequence of entries on the k-th superdiagonal. It satisfies the first-order recurrence equation d_k(n) = n*(n+k)*d_k(n-1) + 1 with d_k(0) = 1 and also the second-order recurrence d_k(n) = (n*(n+k)+1)*d_k(n-1) - (n-1)*(n-1+k)*d_k(n-2) with initial conditions d_k(0) = 1 and d_k(1) = k+2. This latter recurrence is also satisfied by the sequence n!*(n+k)!. From this observation we obtain the finite continued fraction expansion d_k(n) = n!*(n+k)!*(1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) ))))).
Taking the limit as n -> infinity produces a continued fraction representation for the modified Bessel function value BesselI(k,2) = Sum_{i=0..inf} 1/(i!*(i+k)!) = 1/(k! - k!/((k+2) -(k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See A070910 for the case k = 0 and A096789 for the case k = 1. (End)

Examples

			1, 1,  1,   1,    1,     1,
1, 2,  3,   4,    5,     6,
1, 3,  9,  19,   33,    51,
1, 4, 19,  82,  229,   496,
1, 5, 33, 229, 1313,  4581,
1, 6, 51, 496, 4581, 32826,

Links

Crossrefs

Rows include A000012, A000027, A058331. Main diagonal is A006040. Antidiagonal sums are in A099598. Cf. A099599.
Cf. A088699. A228229 (main super and subdiagonal).

Programs

  • Maple
    #A099597
T := proc(n,k) option remember;
if n = 0 then 1 elif k = 0 then 1
else n*k*thisproc(n-1,k-1) + 1
fi
end:
# Diplay entries by antidiagonals
seq(seq(T(n-k,k), k = 0..n), n = 0..10);
# Peter Bala, Aug 19 2013
  • Mathematica
    T[, 0] = T[0, ] = 1;
T[n_, k_] := T[n, k] = n k T[n - 1, k - 1] + 1;
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

Formula

T(n,k) = Sum_{i=0..min(n,k)} C(n,i)*C(k,i)*i!^2. The LDU factorization of this square array is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!^2, 1!^2, 2!^2, ... ). Compare with A088699. - Peter Bala, Nov 06 2007
Recurrence equation: T(n,k) = n*k*T(n-1,k-1) + 1 with boundary conditions T(n,0) = T(0,n ) = 1.
Main subdiagonal and main superdiagonal [1, 3, 19, 229, ...] is A228229. - Peter Bala, Aug 19 2013
nth row/column o.g.f.: HypergeometricPFQ[{1,1,-n},{},x/(x-1)]/(1-x) (see comment in A099599). - Natalia L. Skirrow, Jul 18 2025

A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

Original entry on oeis.org

0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801
Offset: 0

Views

Author

Groux Roland, Jan 17 2011

Keywords

Comments

Integral_{x=0..1} x^n*BesselI(0,2*x^(1/2)) dx = A006040(n)*BesselI(1,2) - a(n)*BesselI(0,2). An elementary consequence is the irrationality of BesselI(0,2)/BesselI(1,2).

Links

Crossrefs

Cf. A006040, A007526, A066998, A228229.

Programs

  • Mathematica
    FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)
  • Maxima
    a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */
  • PARI
    a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", "));  /* show terms */

Formula

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} k/(k!)^2.
a(n) = n * A228229(n-1) for n > 0. (End)

A228230 Recurrence a(n) = (1/2)*n*(n+1)*a(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 2, 7, 43, 431, 6466, 135787, 3802037, 136873333, 6159299986, 338761499231, 22358258949247, 1743944198041267, 158698922021755298, 16663386812284306291, 1999606417474116754921, 271946472776479878669257, 41607810334801421436396322, 7114935567251043065623771063
Offset: 0

Views

Author

Peter Bala, Aug 19 2013

Keywords

Comments

Cf. A006040 and A228229.

Links

Crossrefs

Cf. A006040, A228229.

Programs

  • Maple
    #A228230
a:=proc(n) option remember
if n = 0 then 1 else 1/2n(n+1)thisproc(n-1) + 1
fi
end:
seq(a(n), n = 0..20);

Formula

a(n) = (1/2^n)*n!*(n + 1)!*Sum_{k = 0..n} 2^k/(k!*(k + 1)!).
a(n) = n!*(n+1)!*(the coefficient of x^n*y^(n+1) in the expansion of exp(x + y)/(1 - x*y/2)).
G.f.: (1/(1 - x/2))*(1/sqrt(x))*BesselI(1, 2*sqrt(x)) = Sum_{n >= 0} a(n)*x^n/(n!*(n + 1)!).
Defining recurrence equation: a(n) = (1/2)*n*(n + 1)*a(n-1) + 1 with a(0) = 1.
Alternative recurrence equation: a(0) = 1, a(1) = 2, and for n >= 2, a(n) = ((1/2)*n*(n+1) + 1)*a(n-1) - (1/2)*n*(n - 1)*a(n-2).
The sequence b(n) := (1/2^n)*n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. It follows that we have the finite continued fraction expansion a(n) = (1/2^n)* n!*(n + 1)!*(1 + 1/(1 - 1/(4 - 3/(7 - ... - 1/2*n*(n - 1)/(1/2*n*(n + 1) + 1))))). Taking the limit yields the continued fraction expansion (1/sqrt(2))*BesselI(1,2*sqrt(2)) = Sum_{k >= 0} 2^k/(k!*(k + 1)!) = 1 + 1/(1 - 1/(4 - 3/(7 - 6/(11 - ... - (1/2)*n*(n - 1)/((1/2)*n*(n + 1) + 1 - ...))))) = 2.394833097....

Extensions

Typo in the first formula corrected by Vaclav Kotesovec, Jul 02 2015

A368775 a(n) = (n+1) * (n!)^3 * Sum_{k=0..n} 1/((k+1) * (k!)^3).

Original entry on oeis.org

1, 3, 37, 1333, 106641, 15996151, 4031030053, 1580163780777, 910174337727553, 737241213559317931, 810965334915249724101, 1177521666296942599394653, 2204320559307876546066790417, 5215422443322435907994026126623
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2024

Keywords

Crossrefs

Cf. A217284, A368776.
Cf. A228229, A368771.

Programs

  • PARI
    a(n) = (n+1)*n!^3*sum(k=0, n, 1/((k+1)*k!^3));

Formula

a(n) = (n+1) * n^2 * a(n-1) + 1.

A368837 a(n) = n! * (n+2)! * Sum_{k=0..n} 1/(k! * (k+2)!).

Original entry on oeis.org

1, 4, 33, 496, 11905, 416676, 20000449, 1260028288, 100802263041, 9979424041060, 1197530884927201, 171246916544589744, 28769481979491076993, 5610048986000760013636, 1256650972864170243054465, 320445998080363411978888576
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A099597, A006040, A228229, A368838.

Programs

  • PARI
    a(n) = n!*(n+2)!*sum(k=0, n, 1/(k!*(k+2)!));

Formula

a(n) = n*(n+2)*a(n-1) + 1.
a(n) ~ BesselI(2,2) * n! * (n+2)!. - Vaclav Kotesovec, Jan 09 2024

A368838 a(n) = n! * (n+3)! * Sum_{k=0..n} 1/(k! * (k+3)!).

Original entry on oeis.org

1, 5, 51, 919, 25733, 1029321, 55583335, 3890833451, 342393343689, 36978481118413, 4807202545393691, 740309191990628415, 133255654558313114701, 27717176148129127857809, 6596687923254732430158543, 1781105739278777756142806611, 541456144740748437867413209745
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A099597, A006040, A228229, A368837.

Programs

  • PARI
    a(n) = n!*(n+3)!*sum(k=0, n, 1/(k!*(k+3)!));

Formula

a(n) = n*(n+3)*a(n-1) + 1.
a(n) ~ BesselI(3,2) * n! * (n+3)!. - Vaclav Kotesovec, Jan 09 2024

A368839 a(n) = n! * (n+1)! * (n+2)! * Sum_{k=0..n} 1/(k! * (k+1)! * (k+2)!).

Original entry on oeis.org

1, 7, 169, 10141, 1216921, 255553411, 85865946097, 43276436832889, 31159034519680081, 30847444174483280191, 40718626310317929852121, 69873162748505567626239637, 152602987442736159695707367209, 416606155718669715969281112480571
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A000522, A228229.
Cf. A368840.

Programs

  • PARI
    a(n) = n!*(n+1)!*(n+2)!*sum(k=0, n, 1/(k!*(k+1)!*(k+2)!));

Formula

a(n) = n*(n+1)*(n+2)*a(n-1) + 1.

A368853 a(n) = n! * (n+1)! * Sum_{k=0..n} (-1)^k/(k! * (k+1)!).

Original entry on oeis.org

1, 1, 7, 83, 1661, 49829, 2092819, 117197863, 8438246137, 759442152329, 83538636756191, 11027100051817211, 1720227608083484917, 313081424671194254893, 65747099180950793527531, 15779303803428190446607439, 4291970634532467801477223409
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A000166, A368854.
Cf. A228229.

Programs

  • PARI
    a(n) = n!*(n+1)!*sum(k=0, n, (-1)^k/(k!*(k+1)!));

Formula

a(n) = n * (n+1) * a(n-1) + (-1)^n.
a(n) ~ BesselJ(1,2) * n! * (n+1)!. - Vaclav Kotesovec, Jan 08 2024
Showing 1-10 of 12 results. Next

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