A000813
Expansion of (sin x + cos x)/cos 4x.
Original entry on oeis.org
1, 1, 15, 47, 1185, 6241, 230895, 1704527, 83860545, 796079041, 48942778575, 567864586607, 41893214676705, 574448847467041, 49441928730798255, 782259922208550287, 76946148390480577665, 1379749466246228538241
Offset: 0
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p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end: A000813 := n -> -(-1)^iquo(n,2)*Re(p(n));
seq(A000813(i),i=0..11); # Peter Luschny, Apr 29 2013
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a[n_] := 2*(-1)^Floor[n/2]*Im[Sum[4^j*Binomial[n, j]*PolyLog[-j, I], {j, 0, n}]]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 30 2013, after Peter Luschny *)
With[{nn=20},CoefficientList[Series[(Sin[x]+Cos[x])/Cos[4x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 12 2013 *)
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x='x+O('x^66); Vec(serlaplace((sin(x)+cos(x))/cos(4*x))) \\ Joerg Arndt, Apr 30 2013
A377666
Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) *(2*n)^j.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, 0, 1, 1, -3, -5, 11, 5, 1, 1, -4, -7, 46, 57, 0, 1, 1, -5, -9, 117, 205, -361, -61, 1, 1, -6, -11, 236, 497, -3362, -2763, 0, 1, 1, -7, -13, 415, 981, -15123, -22265, 24611, 1385, 1
Offset: 0
Array A(n, k) starts:
[0] 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 1, 0, -1, 0, 5, 0, -61, ... A122045
[2] 1, -1, -3, 11, 57, -361, -2763, ... A212435
[3] 1, -2, -5, 46, 205, -3362, -22265, ... A225147
[4] 1, -3, -7, 117, 497, -15123, -95767, ... A156201
[5] 1, -4, -9, 236, 981, -47524, -295029, ... A377665
[6] 1, -5, -11, 415, 1705, -120125, -737891, ...
[7] 1, -6, -13, 666, 2717, -262086, -1599793, ...
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GHZeta := (k, n, m) -> m^(k+1)*Zeta(0, -k, 1/(m*n)):
A := (n, k) -> ifelse(n = 0, 1, n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2))):
for n from 0 to 7 do lprint(seq(A(n, k), k = 0..7)) od;
# Alternative:
P := proc(n, k) local j; 2*I*(1 + add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:
A := n -> Im(P(n, k)): seq(lprint(seq(A(n, k), k = 0..7)), n = 0..7);
# Computing the transpose using polynomials P from A363393.
P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1)/(n + 1):
Column := (k, n) -> subs(x = -n, P(k)):
for k from 0 to 6 do seq(Column(k, n), n = 0..9) od;
# According to the definition:
A := (n, k) -> local j; add(binomial(k, j)*euler(j, 0)*(2*n)^j, j = 0..k):
seq(lprint(seq(A(n, k), k = 0..6)), n = 0..7);
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A[n_, k_] := n^k (4^(k+1) HurwitzZeta[-k, 1/(4n)] - 2^(k + 1) HurwitzZeta[-k, 1/(2n)]);
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from mpmath import *
mp.dps = 32; mp.pretty = True
def T(n, k):
p = 2*I*(1+sum(binomial(k, j)*polylog(-j, I)*n^j for j in range(k+1)))
return int(imag(p))
for n in range(8): print([T(n, k) for k in range(7)])
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