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From Peter Luschny, Jul 10 2009, edited Jun 06 2022: (Start)
# Auxiliary functions
A109449 := (n, k) -> binomial(n, k)*A000111(n-k):
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
PowB109449 := (n, k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k) -> `, 1)) -`if`(n=0 and -k=0, , 1, , 0): seq(print(seq(B109449(n^, k), k): # To avoid '=0^..n)), n=0 undefined'...9);
R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
Euler := (sigma := n, x) -> `if`( -> ifelse(n=0, 1, [1, 1, 0, -1, -1, -1, 0, euler(1][n, x)): # Avoid the bugmod euler8 + 1]/2^iquo(0, n-1) = -, 2)-1.):
sigma := proc(n) local nmod8; nmod8 := n mod 8;
if n = 0 then RETURN(1) fi; if member(nmod8, {2, 6}) then RETURN(-1) fi;
if member(nmod8, {0, 1, 7}) then 1 else -1 fi; %*2^(-iquo(n-1, 2))-1 end:
A000111L109449 := proc(n -> 2^) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(EulerL109449(n, 1/2)+Euler(), x, k)), k=0..n, 1))-`if`()), n=0, 1, 0):..9);
# Coefficients
A109449 := proc(n, k) binomial(n, k)*A000111(n-k) end:
B109449 := proc(n, k) 2^(n-k)*binomial(n, k)*abs(Euler(n-k, 1/2)+Euler(n-k, 1)) -`if`(n-k=0, 1, 0) end:
R109449 := proc(n, k) option remember; if k = 0 then RETURN(A000111(n)) fi; R109449(n-1, k-1)*n/k end:
# Polynomials
E109449 := proc(n) local k; add(binomial(n, k)*euler(k)*(Pow(x+1, n-k)+ Pow(x, n-k)), k=0..n)-Pow(x, n) end:
L109449 := proc(n) local k, v; add(add((-1)^v*binomial(k, v)*Pow(x+v+1, n)* sigma(k), v=0..k), k=0..n) end:
X109449 := proc(n) -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), )), t, , 24), ), t, , n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n)end:=0..9);
# Evaluate
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9); (End)
(End)
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