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A256679 A signed triangle of V. I. Arnold for the Springer numbers (A001586).

%I A256679 #35 Mar 01 2025 08:38:50
%S A256679 1,1,0,-2,-3,-3,-8,-6,-3,0,40,48,54,57,57,256,216,168,114,57,0,-1952,
%T A256679 -2208,-2424,-2592,-2706,-2763,-2763,-17408,-15456,-13248,-10824,
%U A256679 -8232,-5526,-2763,0,177280,194688,210144,223392,234216,242448,247974,250737,250737
%N A256679 A signed triangle of V. I. Arnold for the Springer numbers (A001586).
%C A256679 This triangle is denoted R(b) on page 6 of the Arnold reference.
%C A256679 Unsigned version of triangle is triangle of A202818.
%C A256679 First column (m=0) is A000828. - _Robert Israel_, Apr 08 2015
%H A256679 V. I. Arnold, <a href="http://mi.mathnet.ru/eng/umn4470">The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups (in Russian)</a>, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51. (See page 6)
%F A256679 E.g.f.: cosh(x+y)/cosh(2*(x+y))*exp(x).
%F A256679 T(n,m) = Sum_{k=floor(m/2)..floor(n/2)} Sum_{i=0..k} 4^i*E(2*i)*C(2*k,2*i)*C(n-m,2*k-m), where E(n) are the Euler secant numbers A122045.
%e A256679 Triangle begins:
%e A256679     1;
%e A256679     1,   0;
%e A256679    -2,  -3,  -3;
%e A256679    -8,  -6,  -3,   0;
%e A256679    40,  48,  54,  57, 57;
%e A256679   256, 216, 168, 114, 57, 0;
%p A256679 T:= (n,m) -> add(add(4^i*euler(2*i)*binomial(2*k,2*i)*binomial(n-m,2*k-m),i=0..k),k=floor(m/2)..floor(n/2)):
%p A256679 seq(seq(T(n,m),m=0..n),n=0..10); # _Robert Israel_, Apr 08 2015
%p A256679 # Second program, about 100 times faster than the first for the first 100 rows.
%p A256679 Triangle := proc(len) local s, A, n, k;
%p A256679 A := Array(0..len-1,[1]); lprint(A[0]);
%p A256679 for n from 1 to len-1 do
%p A256679    if n mod 2 = 1 then s := 0 else
%p A256679    s := 2^(3*n+1)*(Zeta(0,-n,1/8)-Zeta(0,-n,5/8)) fi;
%p A256679    A[n] := s;
%p A256679    for k from n-1 by -1 to 0 do
%p A256679        s := s + A[k]; A[k] := s od;
%p A256679    lprint(seq(A[k], k=0..n));
%p A256679 od end:
%p A256679 Triangle(100); # _Peter Luschny_, Apr 08 2015
%t A256679 T[n_, m_] := Sum[4^i EulerE[2i] Binomial[2k, 2i] Binomial[n-m, 2k-m], {k, Floor[m/2], n/2}, {i, 0, k}];
%t A256679 Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)
%o A256679 (Maxima)
%o A256679 T(n,m):=(sum((sum(4^i*euler(2*i)*binomial(2*k,2*i),i,0,k))*binomial(n-m,2*k-m),k,floor(m/2),n/2));
%o A256679 (Sage)
%o A256679 def triangle(len):
%o A256679     L = [1]; print(L)
%o A256679     for n in range(1,len):
%o A256679         if is_even(n):
%o A256679             s = 2^(3*n+1)*(hurwitz_zeta(-n,1/8)-hurwitz_zeta(-n,5/8))
%o A256679         else: s = 0
%o A256679         L.append(s)
%o A256679         for k in range(n-1,-1,-1):
%o A256679             s = s + L[k]; L[k] = s
%o A256679         print(L)
%o A256679 triangle(7) # _Peter Luschny_, Apr 08 2015
%Y A256679 Cf. A000828, A001586, A002436, A008280, A122045, A202818, A256665.
%K A256679 sign,tabl
%O A256679 0,4
%A A256679 _Vladimir Kruchinin_, Apr 07 2015