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.

A087208 Expansion of e.g.f. exp(x)/(1-x^2).

Original entry on oeis.org

1, 1, 3, 7, 37, 141, 1111, 5923, 62217, 426457, 5599531, 46910271, 739138093, 7318002277, 134523132927, 1536780478171, 32285551902481, 418004290062513, 9879378882159187, 142957467201379447, 3754163975220491061, 60042136224579367741, 1734423756551866870183
Offset: 0

Views

Author

Vladeta Jovovic, Oct 19 2003

Keywords

Links

Crossrefs

Cf. A000166, A000522, A002747, A010050.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[x]/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 11 2017 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} n!/(n-2*k)!.
a(n) = n*(n-1)*a(n-2) + 1. - Vladeta Jovovic, Aug 24 2004
a(n) = (A000522(n) + (-1)^n*A000166(n))/2. - Vladeta Jovovic, Aug 24 2004
a(n) = Sum_{k=0..n} binomial(n, k)*(1+(-1)^k)k!/2. Binomial transform of A010050 (with interpolated zeros). - Paul Barry, Sep 14 2004
a(n) = Sum_{k=0..n} P(n, k)[1, 0, 1, 0, 1, 0, ...](k). - Ross La Haye, Aug 29 2005
a(n) = (1/(2*exp(1))) * (Integral_{t=0..2} t^n*exp(1-abs(1-t)) dt + Integral_{t=0..oo} ((2+t)^n + (-t)^n) * exp(-t) dt). - Groux Roland, Jan 15 2011
E.g.f.: 1/U(0) where U(k) = 1 - x^2/(1 - 1/(1 + x*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 16 2012
If n is even then a(n) ~ n!*(e/2 + 1/(2*e)) = 1.543080634815243... * n!, if n is odd then a(n) ~ n!*(e/2 - 1/(2*e)) = 1.175201193643801... * n!. - Vaclav Kotesovec, Nov 20 2012
Conjecture: a(n) -a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 29 2013
From Peter Bala, Sep 05 2022: (Start)
The e.g.f. A(x) satisfies the differential equation (x^2 - 1)*A'(x) + (1 + 2*x - x^2)*A(x) = 0 with A(0) = 1. Mathar's recurrence above follows from this.
For k a positive integer, reducing the sequence modulo k produces a purely periodic sequence whose period divides k. For example, modulo 5 the sequence becomes [1, 1, 3, 2, 2, 1, 1, 3, 2, 2, ...] of period 5. (End)

Extensions

Definition clarified by Harvey P. Dale, Aug 11 2017

A260325 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

Original entry on oeis.org

1, 2, 1, 5, 2, 2, 16, 9, 6, 6, 65, 28, 12, 24, 24, 326, 185, 140, 60, 120, 120, 1957, 846, 750, 120, 360, 720, 720, 13700, 7777, 2562, 5250, 840, 2520, 5040, 5040, 109601, 47384, 47096, 40656, 1680, 6720, 20160, 40320, 40320, 986410, 559953, 378072, 181944, 365904, 15120, 60480, 181440, 362880, 362880
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2015

Keywords

Examples

			Triangle begins:
     1;
     2,   1;
     5,   2,   2;
    16,   9,   6,   6;
    65,  28,  12,  24,  24;
   326, 185, 140,  60, 120, 120;
  1957, 846, 750, 120, 360, 720, 720;
  ...

Links

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Crossrefs

Rows, column sums give A000522, A002747, A002750, A002751.
Main diagonal gives A000142.
Cf. A260322, A260323, A260324.

Programs

  • Maple
    A260325 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n+1)!,j=1..(r+1)/n) ;
        %*r! ;
    end if;
end proc:
for r from 0 to 20 do
    for n from 1 to r+1 do
        printf("%a,",A260325(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015
  • Mathematica
    T[n_, k_] := Which[n == 0, 1, k > n+1, 0, True, Sum[1/(n-j*k+1)!, {j, 1, (n+1)/k}]*n!];
Table[T[n, k], {n, 0, 9}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Apr 25 2023 *)

A002742 Logarithmic numbers.

Original entry on oeis.org

2, -6, 24, -80, 450, -2142, 17696, -112464, 1232370, -9761510, 132951192, -1258797696, 20476388114, -225380451870, 4261074439680, -53438049741152, 1151146814425506, -16199301256675974, 391615698778725080, -6109914386833902960
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From Peter Bala, Sep 06 2022: (Start)
Conjectures: Let k be a positive integer.
1) for n >= 1, a(n+2*k) - a(n) is divisible by 2*k; if true, then the reduction of the sequence modulo 2*k gives a periodic sequence with period dividing 2*k.
2) for n >= 1, a(n+2*k+1) + a(n) is divisible by 2*k+1; if true, then the reduction of the sequence modulo 2*k+1 gives a periodic sequence with period dividing 4*k + 2. (End)

References

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • 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. A002741.

Programs

  • Mathematica
    Table[(-1)^(n-1)Sum[Binomial[n+1,2k+1](n-2k)/(k+1)(2k+1)!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Dec 16 2017 *)
  • Maxima
    makelist((-1)^(n-1)*sum(binomial(n+1,2*k+1)*(n-2*k)/(k+1)*(2*k+1)!,k,0,n),n,0,12); /* Emanuele Munarini, Dec 16 2017 */
  • PARI
    first(n) = x='x+O('x^(n+1)); Vec(serlaplace((2*x/(1-x^2)+log(1-x^2))*exp(-x))) \\ Iain Fox, Dec 16 2017

Formula

E.g.f.: (2*x/(1-x^2)+log(1-x^2))*exp(-x). - Sean A. Irvine, Aug 11 2014
a(n) = 2*A002747(n) - a(n-1). - R. J. Mathar, Jul 24 2015
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = (-1)^(n-1)*Sum_{k=0..n} binomial(n+1,2*k+1)*((n-2*k)/(k+1))*(2*k+1)!.
a(n+3)+a(n+2)-(n+2)*(n+3)*a(n+1)-(n+2)*(n+3)*a(n) = 2*(-1)^n*(n+3).
(n+3)*a(n+4)+(2*n+7)*a(n+3)-(n+2)*(n+4)^2*a(n+2)-(n+3)*(n+4)*(2*n+5)*a(n+1)-(n+2)*(n+3)*(n+4)*a(n) = 0.
E.g.f.: A(x) = - D(exp(-x)*log(1-x^2)), where D is the derivative with respect to x. (End)
a(n) ~ n! * (exp(-1) - (-1)^n * exp(1)). - Vaclav Kotesovec, Dec 16 2017

Extensions

More terms from Jeffrey Shallit
More terms from Sean A. Irvine, Aug 11 2014

A260324 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

Original entry on oeis.org

1, 0, 1, 1, -2, 2, 2, 9, -6, 6, 9, -28, 12, -24, 24, 44, 185, 100, 60, -120, 120, 265, -846, -690, -120, 360, -720, 720, 1854, 7777, 2478, 5250, -840, 2520, -5040, 5040, 14833, -47384, 33656, -40656, 1680, -6720, 20160, -40320, 40320, 133496, 559953, -347832, 181944, 359856, 15120, -60480, 181440, -362880, 362880
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2015

Keywords

Examples

			Triangle begins:
1,
0,1,
1,-2,2,
2,9,-6,6,
9,-28,12,-24,24,
44,185,100,60,-120,120,
265,-846,-690,-120,360,-720,720,
...

Links

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Crossrefs

Rows, column sums give A000166, A002747, A002748, A002749.
Cf. A260322, A260323, A260325.

Programs

  • Maple
    A260324 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( (-1)^(r-j*n+1)/(r-j*n+1)!,j=1..(r+1)/n) ;
        %*r! ;
    end if;
end proc:
for r from 0 to 20 do
    for n from 1 to r+1 do
        printf("%a,",A260324(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015
  • Mathematica
    T[n_, k_] := If[k == 0, 1, If[n > k + 1, 0, k! Sum[(-x)^(k - j n + 1)/(k - j n + 1)!, {j, 1, (k + 1)/n}]]];
Table[T[n, k] /. x -> 1, {k, 0, 9}, {n, 1, k + 1}] // Flatten (* Jean-François Alcover, Mar 30 2020 *)
Showing 1-4 of 4 results.

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

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