A354326
Expansion of e.g.f. 1/(1 - x/8 * (exp(4 * x) - 1)).
Original entry on oeis.org
1, 0, 1, 6, 38, 280, 2538, 27524, 341912, 4754880, 73322360, 1244282512, 23048700912, 462565343552, 9996300546512, 231444311970720, 5715911385442688, 149988948332148736, 4167328800543910272, 122218355207805620480, 3773036019063284645120
Offset: 0
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With[{nn=20},CoefficientList[Series[1/(1-x/8 (Exp[4x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 03 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/8*(exp(4*x)-1))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*4^(j-2)*binomial(i, j)*v[i-j+1])/2); v;
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a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*k!*stirling(n-k, k, 2)/(2^k*(n-k)!));
A354327
Expansion of e.g.f. 1/(1 + x/4 * log(1 - 2 * x)).
Original entry on oeis.org
1, 0, 1, 3, 22, 180, 1902, 23730, 344872, 5706288, 105960600, 2181449160, 49311653616, 1214109056160, 32339248301808, 926527371653520, 28410493609687680, 928335829570087680, 32201658919855225728, 1181755749910942408320, 45744743939940787150080
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/4*log(1-2*x))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 2^(j-3)/(j-1)*v[i-j+1]/(i-j)!)); v;
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a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);
A354320
Expansion of e.g.f. 1/(1 - 4*x)^(x/8).
Original entry on oeis.org
1, 0, 1, 6, 67, 1020, 19767, 464310, 12802121, 405017928, 14454250785, 574259123790, 25131727031163, 1201109694719220, 62238037299307863, 3475264183358721390, 208017790077615619665, 13286691367919839674000, 901996048369381319539713
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-4*x)^(x/8)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*4^(j-2)/(j-1)*v[i-j+1]/(i-j)!)/2); v;
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a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*abs(stirling(n-k, k, 1))/(2^k*(n-k)!));
Showing 1-3 of 3 results.