A355409
Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).
Original entry on oeis.org
1, 1, 7, 55, 571, 7471, 117307, 2148175, 44958571, 1058555791, 27693129307, 796934764495, 25018548004171, 850870651904911, 31163746960955707, 1222922731101304015, 51189052318085027371, 2276586205163067346831, 107204914362429152404507
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(2*x)-exp(3*x))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-2^j)*binomial(i, j)*v[i-j+1])); v;
A370163
a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.
Original entry on oeis.org
2, 1, 5, 25, 161, 1321, 13025, 149605, 1963841, 29004721, 475975745, 8591917885, 169193833121, 3609452038921, 82924458549665, 2041207822721365, 53594538159184001, 1495143168658285921, 44164021453758342785, 1377005070100813288045, 45193800193226286112481
Offset: 0
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seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*(1 + p)/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 26 2024
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def a(m):
if m==0:
return 2
else:
return (-1)^m+(-2)^m+1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])
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f=2*(1+e^x)/(1+e^x+e^(2*x)-e^(3*x))
print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])
A370456
a(0) = 1, a(n) = (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.
Original entry on oeis.org
1, 2, 6, 29, 192, 1577, 15516, 178229, 2339952, 34559057, 567117876, 10237161629, 201592448712, 4300618438937, 98803485774636, 2432074390036229, 63857242954421472, 1781444969999245217, 52620896463516221796, 1640684857196257578029, 53847865360369426418232
Offset: 0
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seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*p^2/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 23 2024
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def a(m):
if m==0:
return 1
else:
return 1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])
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