A256665 Triangle of Arnold L(b) for Springer numbers.

0, 1, 1, 0, 1, 2, 11, 11, 10, 8, 0, 11, 22, 32, 40, 361, 361, 350, 328, 296, 256, 0, 361, 722, 1072, 1400, 1696, 1952, 24611, 24611, 24250, 23528, 22456, 21056, 19360, 17408, 0, 24611, 49222, 73472, 97000, 119456, 140512, 159872, 177280

Offset

0,6

Comments

Named after Soviet and Russian mathematician Vladimir Igorevich Arnold (1937-2010). - Amiram Eldar, Jun 13 2021

Links

Table of n, a(n) for n=0..44.

Vladimir Igorevich Arnol'd, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., Vol. 47, No. 1 (1992), pp. 3-45; English version, Russian Math. Surveys, Vol. 47 (1992), pp. 1-51.

Formula

E.g.f.: sinh(x+y)/cosh(2*(x+y))*exp(-y).

T(n,m) = abs(Sum_{k=floor((n-m+1)/2)..floor((n+1)/2)} C(m,2*k+m-n-1)*Sum_{i=0..k} 4^i*Euler(2*i)*C(2*k-1,2*i)).

Example

Triangle begins:
    0;
    1,   1;
    0,   1,   2;
   11,  11,  10,   8;
    0,  11,  22,  32,  40;
  361, 361, 350, 328, 296, 256;

Mathematica

T[n_, m_] := Abs[Sum[Binomial[m, 2*k+m-n-1]*Sum[4^i*EulerE[2*i]*Binomial[2*k-1, 2*i], {i, 0, k}], {k, Floor[(n-m+1)/2], (n+1)/2}]]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 07 2015, translated from Maxima *)

Prog

(Maxima)
T(n, m):=abs(sum(binomial(m, 2*k+m-n-1)*sum(4^i*euler(2*i)*binomial(2*k-1, 2*i), i, 0, k), k, floor((n-m+1)/2), (n+1)/2));

Crossrefs

Sequence in context: A153521 A153650 A338049 * A086862 A345392 A027828

Adjacent sequences: A256662 A256663 A256664 * A256666 A256667 A256668

Keyword

nonn,tabl,changed

Author

Vladimir Kruchinin, Apr 07 2015

Status

approved