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A256679
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A signed triangle of V. I. Arnold for the Springer numbers (A001586).
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0
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1, 1, 0, -2, -3, -3, -8, -6, -3, 0, 40, 48, 54, 57, 57, 256, 216, 168, 114, 57, 0, -1952, -2208, -2424, -2592, -2706, -2763, -2763, -17408, -15456, -13248, -10824, -8232, -5526, -2763, 0, 177280, 194688, 210144, 223392, 234216, 242448, 247974, 250737, 250737
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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This triangle is denoted R(b) on page 6 of the Arnold reference.
Unsigned version of triangle is triangle of A202818.
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LINKS
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FORMULA
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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.
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EXAMPLE
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1;
1, 0;
-2, -3, -3;
-8, -6, -3, 0;
40, 48, 54, 57, 57;
256, 216, 168, 114, 57, 0;
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MAPLE
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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)):
# 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:
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MATHEMATICA
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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}];
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PROG
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(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)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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