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

Author

Vladimir Kruchinin, Apr 07 2015

Keywords

• nonn
• tabl

Comments

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

Examples

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

Links

• 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.

Programs

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 *)

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));

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)).