A185323
E.g.f. A(x) = 1/(2-tan(x)-sec(x)).
Original entry on oeis.org
1, 1, 3, 14, 87, 676, 6303, 68564, 852387, 11921476, 185259603, 3166825364, 59054916687, 1193026564276, 25955467164903, 605021502144164, 15043243752072987, 397412126087559076, 11116403953041202203, 328222705791221254964
Offset: 0
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T:= proc(n, k) option remember;
if k=n then 1
elif k<0 or k>n then 0
else T(n-1, k-1) +k*T(n-1, k) +k*(k+1)/2 *T(n-1, k+1)
fi
end:
a:= n-> add(k! * T(n, k), k=0..n):
seq(a(n), n=0..30); # Alois P. Heinz, Feb 18 2011
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CoefficientList[Series[1/(2-Tan[x]-Sec[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)
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x = 'x + O('x^30); Vec(serlaplace(1/(2-tan(x)-1/cos(x)))) \\ Michel Marcus, Jun 27 2017
A185324
E.g.f. log(1/(2-tan(x)-sec(x))).
Original entry on oeis.org
0, 1, 2, 7, 34, 215, 1682, 15727, 171274, 2130275, 29799722, 463123747, 7916886514, 147635940335, 2982555226562, 64888568231767, 1512552803481754, 37608099684426395, 993530210286226202, 27791008680163167787, 820556749933610580994, 25502885614554196884455
Offset: 0
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T:= proc(n,k) option remember;
if k=n then 1
elif k<0 or k>n then 0
else T(n-1, k-1) +k*T(n-1,k) +k*(k+1)/2 *T(n-1, k+1)
fi
end:
a:= n-> add((k-1)! * T(n,k), k=1..n):
seq(a(n), n=0..20); # Alois P. Heinz, Feb 17 2011
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T[n_, k_] := T[n, k] = If[k==n, 1, If[k<0 || k>n, 0, T[n-1, k-1] + k*T[n-1, k] + k*(k+1)/2*T[n-1, k+1]]]; a[n_] := Sum[(k-1)!*T[n, k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 03 2015, after Alois P. Heinz *)
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a[0]:0$a[1]:1$
a[n]:=sum((-1)^floor(p/2)*(mod(p+1,2)-(-1)^p*4^floor(p/2))*binomial(n-1,p)*a[n-p],p,1,n-1)-mod(n-1,2)*(%i)^n;
makelist(a[n],n,0,100); /* Tani Akinari, Oct 30 2017 */
A240561
The main diagonal in the difference table of A240559.
Original entry on oeis.org
0, 1, -10, 178, -5296, 238816, -15214480, 1301989648, -144118832896, 20040052293376, -3419989086092800, 702831038438522368, -171209091176316215296, 48783404012394865985536, -16074763418934659189278720, 6065554251200571899397081088, -2598468976240882751482797162496
Offset: 0
a(n) is the main diagonal in this difference table D(n, k):
[ 0, 0, 1, -3, -5, 45, 61, -1113, -1385]
[ 0, 1, -2, -8, 40, 106, -1052, -2498]
[ 1, -1, -10, 32, 146, -946, -3550]
[ 0, -11, 22, 178, -800, -4496]
[ -11, 11, 200, -622, -5296]
[ 0, 211, -422, -5918]
[ 211, -211, -6340]
[ 0, -6551]
[-6551]
D(n, 0) = A240560(n).
D(0, n) = A240559(n).
D(2*n, 0) = (-1)^(n+1)*A147315(2*n, 2).
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A240561_list := proc(len) local A, m, n, k;
n := 2*len-1; A := array(0..n, 0..n);
for m from 0 to n do
A[m, 0] := euler(m) + 2^(m+1)*euler(m+1,0);
for k from m-1 by -1 to 0 do
A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1]
od od; [seq(A[k, k], k=0..len-1)] end:
A240561_list(17);
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Table[-Sum[Binomial[n, k]*EulerE[n+k+1], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Apr 06 2015 *)
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a(n):=-sum(binomial(n,k)*euler(n+k+1),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */