A298213
Triangle read by rows, expansion of exp(x*exp(z)*tan(z)).
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 12, 32, 12, 1, 0, 41, 160, 110, 20, 1, 0, 142, 856, 900, 280, 30, 1, 0, 685, 4816, 7231, 3360, 595, 42, 1, 0, 3192, 29952, 58632, 37856, 9800, 1120, 56, 1, 0, 19921, 199680, 493100, 416640, 147126, 24192, 1932, 72, 1
Offset: 0
Triangle starts:
0: 1;
1: 0, 1;
2: 0, 2, 1;
3: 0, 5, 6, 1;
4: 0, 12, 32, 12, 1;
5: 0, 41, 160, 110, 20, 1;
6: 0, 142, 856, 900, 280, 30, 1;
7: 0, 685, 4816, 7231, 3360, 595, 42, 1;
8: 0, 3192, 29952, 58632, 37856, 9800, 1120, 56, 1;
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gf := exp(x*exp(z)*tan(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A298213_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
seq(A298213_row(n), n=0..8);
A305710
Expansion of e.g.f. exp(sec(x)*exp(x) - 1).
Original entry on oeis.org
1, 1, 3, 11, 53, 297, 1959, 14499, 120409, 1097025, 10931771, 117685163, 1363889133, 16887554569, 222672557631, 3110742121059, 45912214062961, 713290136581697, 11636755988405555, 198800967493444875, 3549276499518132325, 66076184834921382313, 1280502976522048458647
Offset: 0
exp(sec(x)*exp(x) - 1) = 1 + x + 3*x^2/2! + 11*x^3/3! + 53*x^4/4! + 297*x^5/5! + 1959*x^6/6! + 14499*x^7/7! + ...
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a:=series(exp(sec(x)*exp(x)-1),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Exp[Sec[x] Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(2 I)^k EulerE[k, 1/2 - I/2] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Showing 1-2 of 2 results.