OFFSET
0,4
COMMENTS
This triangle is denoted R(b) on page 6 of the Arnold reference.
Unsigned version of triangle is triangle of A202818.
First column (m=0) is A000828. - Robert Israel, Apr 08 2015
LINKS
V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups (in Russian), Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51. (See page 6)
FORMULA
E.g.f.: cosh(x+y)/cosh(2*(x+y))*exp(x).
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.
EXAMPLE
1;
1, 0;
-2, -3, -3;
-8, -6, -3, 0;
40, 48, 54, 57, 57;
256, 216, 168, 114, 57, 0;
MAPLE
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)):
seq(seq(T(n, m), m=0..n), n=0..10); # Robert Israel, Apr 08 2015
# Second program, about 100 times faster than the first for the first 100 rows.
Triangle := proc(len) local s, A, n, k;
A := Array(0..len-1, [1]); lprint(A[0]);
for n from 1 to len-1 do
if n mod 2 = 1 then s := 0 else
s := 2^(3*n+1)*(Zeta(0, -n, 1/8)-Zeta(0, -n, 5/8)) fi;
A[n] := s;
for k from n-1 by -1 to 0 do
s := s + A[k]; A[k] := s od;
lprint(seq(A[k], k=0..n));
od end:
Triangle(100); # Peter Luschny, Apr 08 2015
MATHEMATICA
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}];
Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
PROG
(Maxima)
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));
(Sage)
def triangle(len):
L = [1]; print(L)
for n in range(1, len):
if is_even(n):
s = 2^(3*n+1)*(hurwitz_zeta(-n, 1/8)-hurwitz_zeta(-n, 5/8))
else: s = 0
L.append(s)
for k in range(n-1, -1, -1):
s = s + L[k]; L[k] = s
print(L)
triangle(7) # Peter Luschny, Apr 08 2015
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Vladimir Kruchinin, Apr 07 2015
STATUS
approved