A307733
a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).
Original entry on oeis.org
1, 1, 4, 14, 54, 220, 934, 4090, 18344, 83850, 389214, 1829736, 8693962, 41685714, 201442188, 980091814, 4797070022, 23603701828, 116688837886, 579312087802, 2887020896016, 14437318756818, 72424982972862, 364366674463824, 1837954750285458
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 2 x^3 + x^4])/(2 x), {x, 0, nmax}], x]
A349185
G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^2 * A(x)).
Original entry on oeis.org
1, 1, 4, 11, 35, 111, 365, 1221, 4160, 14371, 50251, 177503, 632514, 2271027, 8208259, 29840993, 109049568, 400352639, 1475929092, 5461571729, 20279092033, 75531360153, 282123848574, 1056539226257, 3966214054639, 14922195004703, 56258116929483, 212505815364639, 804142811583006
Offset: 0
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nmax = 28; A[] = 0; Do[A[x] = (1 - x)/(1 - 2 x - x^2 - x^2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 28; CoefficientList[Series[(1 - 2 x - x^2 - Sqrt[1 - 4 x - 2 x^2 + 8 x^3 + x^4])/(2 x^2), {x, 0, nmax}], x]
a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]
A376574
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)/(1 - x^3)).
Original entry on oeis.org
1, 1, 2, 5, 15, 46, 147, 486, 1646, 5684, 19940, 70864, 254592, 923153, 3374046, 12417246, 45975677, 171141378, 640105278, 2404375805, 9066188052, 34305301482, 130219435385, 495735347502, 1892254721982, 7240580768021, 27768359445128, 106718055778871
Offset: 0
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A376574 := proc(n)
add(A000108(n-3*k)*binomial(n-2*k-1,k),k=0..floor(n/3)) ;
end proc:
seq(A376574(n),n=0..80) ;
# R. J. Mathar, Oct 24 2024
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a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
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my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x^3))))
Showing 1-3 of 3 results.