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.

A008969 Triangle of differences of reciprocals of unity.

Original entry on oeis.org

1, 1, 3, 1, 11, 7, 1, 50, 85, 15, 1, 274, 1660, 575, 31, 1, 1764, 48076, 46760, 3661, 63, 1, 13068, 1942416, 6998824, 1217776, 22631, 127, 1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255, 1, 1026576, 7245893376, 673781602752, 1413470290176, 117550462624, 747497920, 833375, 511
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  1,      3;
  1,     11,         7;
  1,     50,        85,         15;
  1,    274,      1660,        575,        31;
  1,   1764,     48076,      46760,      3661,       63;
  1,  13068,   1942416,    6998824,   1217776,    22631,    127;
  1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
  ...

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.

Links

Crossrefs

Cf. A001236, A001237, A001238, A001240, A001241, A001242.
Columns include A000254, A000424, A001236, A001237, A001238. Right-hand columns include A000225, A001240, A001241, A001242.

Programs

  • Maple
    T:= (n,k)-> `if`(k<=n, (n-k+2)!^k *
     add((-1)^(j+1)*binomial(n-k+2, j)/ j^k, j=1..n-k+2), 0):
seq(seq(T(n,k), k=0..n), n=0..7); # Alois P. Heinz, Sep 05 2008
  • Mathematica
    T[n_, k_] := If[k <= n, (n-k+2)!^k*Sum[(-1)^(j+1)*Binomial[n-k+2, j]/j^k, {j, 1, n-k+2}], 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

A001236 Differences of reciprocals of unity.

Original entry on oeis.org

15, 575, 46760, 6998824, 1744835904, 673781602752, 381495483224064, 303443622431870976, 327643295527342080000, 466962174913357393920000, 858175477913267353681920000, 1993920215002599923346309120000, 5758788816015998806424467537920000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
  • 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

Column 3 in triangle A008969.
Cf. A000254, A000424, A001237, A001238.

Programs

  • Maple
    a:= n-> (n+1)!^3* add((-1)^(k+1) *binomial(n+1, k)/ k^3, k=1..n+1):
seq (a(n), n=1..15);  # Alois P. Heinz, Sep 05 2008
  • Mathematica
    h = HarmonicNumber; a[n_] := ((n+1)!^3/6)*(h[n+1, 1]^3 + 3*h[n+1, 1]*h[n+1, 2] + 2*h[n+1, 3]); Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Feb 26 2015, after Vladeta Jovovic *)

Formula

a(n) = (n+1)!^3 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} 1/(i*j*k).
From Vladeta Jovovic, Jan 30 2005: (Start)
a(n) = (n!^3/6)*(H(n, 1)^3+3*H(n, 1)*H(n, 2)+2*H(n, 3)), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
a(n) = (n!^3/6)*((Psi(n+1)+gamma)^3+3*(Psi(n+1)+gamma)*(-Psi(1, n+1)+1/6*Pi^2)+Psi(2, n+1)+2*Zeta(3)).
a(n) = n!^3*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^3.
Sum_{n>=0} a(n)*x^n/n!^3 = polylog(3, x/(x-1))/(x-1). (offset 2). (End)

Extensions

More terms from Alois P. Heinz, Sep 05 2008

A060237 a(n) = n!^2 * Sum_{m=1..n}( Sum_{k=1..m} 1/(k*m) ).

Original entry on oeis.org

1, 7, 85, 1660, 48076, 1942416, 104587344, 7245893376, 628308907776, 66687811660800, 8506654697548800, 1284292319599411200, 226530955276874956800, 46165213716463676620800
Offset: 1

Views

Author

Leroy Quet, Mar 21 2001

Keywords

Examples

			a(2) = 2!^2 *(1/(1*1) + 1/(1*2) + 1/(2*2)) = 7.

Links

Crossrefs

Essentially the same as A000424.

Programs

  • Magma
    [Factorial(n)^2*(&+[(-1)^(k+1)*Binomial(n,k)/k^2: k in [1..n]]): n in [1..30]]; // G. C. Greubel, Aug 30 2018
  • Mathematica
    Table[n!^2*Sum[(-1)^(k+1)*Binomial[n,k]/k^2, {k,1,n}], {n,1,30}] (* or *) Table[n!^2*Sum[Sum[1/(k*m), {k,1,m}], {m,1,n}], {n,1,30}](* G. C. Greubel, Aug 30 2018 *)
  • PARI
    for(n=1,30, print1(n!^2*sum(k=1,n, (-1)^(k+1)*binomial(n,k)/k^2), ", ")) \\ G. C. Greubel, Aug 30 2018

Formula

a(n) = a(n-1) * n^2 + (n-1)! *n! * Sum_{k=1..n} 1/k.
From Vladeta Jovovic, Jan 29 2005: (Start)
Sum_{n>=0} a(n)*x^n/n!^2 = -dilog(1/(1-x))/(1-x).
a(n) = n!^2*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^2. (End)
From Vaclav Kotesovec, Oct 23 2017: (Start)
a(n) = (3*n^2 - 3*n + 1)*a(n-1) - 3*(n-1)^4*a(n-2) + (n-2)^3*(n-1)^3*a(n-3).
a(n) ~ n!^2 * log(n)^2/2 * (1 + 2*gamma/log(n) + (Pi^2/6 + gamma^2)/log(n)^2), where gamma is the Euler-Mascheroni constant (A001620). (End)

A294117 a(n) = (n!)^2 * Sum_{k=1..n} binomial(n,k) / k^2.

Original entry on oeis.org

1, 9, 139, 3460, 129076, 6831216, 492314544, 46810296576, 5724123883776, 881047053849600, 167511790501401600, 38685942660873830400, 10689310289146278297600, 3485920800452969462169600, 1325434521073620201431040000, 581241452210335678204477440000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 23 2017

Keywords

Links

Crossrefs

Cf. A000424, A060237, A103213.

Programs

  • Magma
    I:=[1,9,139,3460]; [n le 4 select I[n] else  (5*n^2- 7*n+3)*Self(n-1)-(n-1)^2*(9*n^2-24*n+17)*Self(n-2)+(n-2)^3*(n-1)^2*(7*n-13)*Self(n-3)-2*(n-3)^3*(n-2)^3*(n-1)^2*Self(n-4): n in [1..16]]; // Vincenzo Librandi, Oct 24 2017
  • Maple
    f:= gfun:-rectoproc({a(n) = (5*n^2 - 7*n + 3)*a(n-1) - (n-1)^2*(9*n^2 - 24*n + 17)*a(n-2) + (n-2)^3*(n-1)^2*(7*n - 13)*a(n-3) - 2*(n-3)^3*(n-2)^3*(n-1)^2*a(n-4),a(1)=1,a(2)=9,a(3)=139,a(4)=3460},a(n),remember):
map(f, [$1..20]); # Robert Israel, Oct 23 2017
  • Mathematica
    Table[n!^2*Sum[Binomial[n, k]/k^2, {k, 1, n}], {n, 1, 20}]
Table[n!^2*n*HypergeometricPFQ[{1, 1, 1, 1 - n}, {2, 2, 2}, -1], {n, 1, 20}]

Formula

a(n) = (5*n^2 - 7*n + 3)*a(n-1) - (n-1)^2*(9*n^2 - 24*n + 17)*a(n-2) + (n-2)^3*(n-1)^2*(7*n - 13)*a(n-3) - 2*(n-3)^3*(n-2)^3*(n-1)^2*a(n-4).
a(n) ~ (n!)^2 * 2^(n+2) / n^2.
Showing 1-4 of 4 results.

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

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