A002456 Joffe's central differences of 0, A241171(n,n-1).

0, 1, 30, 1260, 75600, 6237000, 681080400, 95351256000, 16672848192000, 3563821301040000, 914714133933600000, 277707211062240960000, 98459829376612704000000, 40319300129722902288000000, 18888041368462498071840000000, 10037644841525784689606400000000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
• 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. 112.
• S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
• 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

• Vaclav Kotesovec, Table of n, a(n) for n = 1..235

Crossrefs

A diagonal of A241171.

Programs

Maple

T:=proc(n,k) option remember;
if k = 0 or k > n then 0
elif k=1 then 1
else k*(2*k-1)*T(n-1,k-1)+k^2*T(n-1,k); fi;
end;
[seq(T(n,n-1),n=1..30)];

Mathematica

T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, _] = 0;
a[n_] := T[n, n-1];
Array[a, 16] (* Jean-François Alcover, Jul 03 2019 *)

Formula

a(n) ~ sqrt(Pi) * 2^n * n^(2*n+3/2) / (3 * exp(2*n)). - Vaclav Kotesovec, Apr 25 2014

Extensions

Entry revised by N. J. A. Sloane, Apr 22 2014