A060237 -id:A060237 - cp's OEIS frontend

A000424 Differences of reciprocals of unity.

7, 85, 1660, 48076, 1942416, 104587344, 7245893376, 628308907776, 66687811660800, 8506654697548800, 1284292319599411200, 226530955276874956800, 46165213716463676620800, 10765453901922078105600000, 2848453606917036402278400000, 848800150518516674081587200000

Offset: 1

N. J. A. Sloane

nonn

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).

Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Essentially the same as A060237.

Column 2 in triangle A008969.

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]; a[n_] := T[n+1, 2]; Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A008969 *)

From Vaclav Kotesovec, Oct 23 2017: (Start)
a(n) = (3*n^2 + 3*n + 1)*a(n-1) - 3*n^4*a(n-2) + (n-1)^3*n^3*a(n-3).
a(n) ~ Pi * log(n)^2 * n^(2*n + 3) * (1 + 2*gamma/log(n) + (gamma^2 + Pi^2/6) / log(n)^2) / exp(2*n), where gamma is the Euler-Mascheroni constant (A001620). (End)

More terms from Vaclav Kotesovec, Oct 23 2017

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

Vaclav Kotesovec, Oct 23 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..238

Cf. A000424, A060237, A103213.

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

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

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}]

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.

cp's OEIS Frontend

A000424 Differences of reciprocals of unity.

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