A007347 Maximal Eulerian numbers of second kind. (Formerly M1889)

1, 1, 2, 8, 58, 444, 4400, 58140, 785304, 12440064, 238904904, 4642163952, 101180433024, 2549865473424, 64728375139872, 1797171220690560, 56071264983487776, 1758073054805500608, 59321137058404865280, 2206689692993315764416, 82380712138316751438720

Offset

0,3

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 256.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..404

R. G. Wilson, V, Letter to N. J. A. Sloane, Apr. 1994

Formula

a(n) = max_{k=0..n} A201637(n, k). - Sean A. Irvine, Dec 28 2017

Maple

a:= n-> max(seq(combinat[eulerian2](n, k), k=0..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 28 2017

Mathematica

Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n-1, k] (k+1) + Eulerian2[n-1, k-1] (2n-k-1)]];
a[n_] := Max[Table[Eulerian2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2019 *)

Prog

(Python)
from sympy.core.cache import cacheit
@cacheit
def eulerian2(n, k): return 1 if k==0 else 0 if k==n else eulerian2(n - 1, k)*(k + 1) + eulerian2(n - 1, k - 1)*(2*n - k - 1)
def a(n): return max(eulerian2(n, k) for k in range(n+1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Dec 29 2017
(PARI) T(n, m) = if ((n==0) || (m==0), 1, sum(k=0, n-m-1, (-1)^(n+k)* binomial(2*n+1, k)*stirling(2*n-m-k, n-m-k, 1)));
a(n) = if (n==0, 1, vecmax(vector(n+1, m, T(n, m-1)))); \\ Michel Marcus, May 29 2019

Crossrefs

Cf. A201637.

Sequence in context: A334617 A191481 A229529 * A027257 A308352 A185898

Adjacent sequences: A007344 A007345 A007346 * A007348 A007349 A007350

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Extensions

More terms from Sean A. Irvine, Dec 28 2017

Status

approved