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Triangle of Arnold L(b) for Springer numbers.
1

%I #15 Jun 13 2021 07:12:14

%S 0,1,1,0,1,2,11,11,10,8,0,11,22,32,40,361,361,350,328,296,256,0,361,

%T 722,1072,1400,1696,1952,24611,24611,24250,23528,22456,21056,19360,

%U 17408,0,24611,49222,73472,97000,119456,140512,159872,177280

%N Triangle of Arnold L(b) for Springer numbers.

%C Named after Soviet and Russian mathematician Vladimir Igorevich Arnold (1937-2010). - _Amiram Eldar_, Jun 13 2021

%H Vladimir Igorevich Arnol'd, <a href="http://mi.mathnet.ru/eng/umn4470">The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups</a>, Uspekhi Mat. nauk., Vol. 47, No. 1 (1992), pp. 3-45; <a href="http://iopscience.iop.org/article/10.1070/RM1992v047n01ABEH000861/pdf">English version</a>, Russian Math. Surveys, Vol. 47 (1992), pp. 1-51.

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

%F T(n,m) = abs(Sum_{ k=floor((n-m+1)/2)..(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)).

%e 0;

%e 1, 1;

%e 0, 1, 2;

%e 11, 11, 10, 8;

%e 0, 11, 22, 32, 40;

%e 361, 361, 350, 328, 296, 256;

%t 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 *)

%o (Maxima)

%o 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));

%K nonn,tabl

%O 0,6

%A _Vladimir Kruchinin_, Apr 07 2015