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A352879
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3*k+1) * a(k).
1
1, 1, 2, 3, 5, 10, 21, 44, 94, 207, 463, 1034, 2289, 5023, 10952, 23784, 51550, 111707, 242370, 527176, 1150643, 2522064, 5553977, 12290257, 27326784, 61031999, 136860175, 307979631, 695115874, 1572737222, 3565447689, 8095730133, 18405209484
OFFSET
0,3
FORMULA
G.f. A(x) satisfies: A(x) = 1 + x * A(x^3/(1 - x)^3) / (1 - x)^2.
E.g.f.: 1 + exp(x) * Sum_{n>=0} a(n) * x^(3*n+1) / (3*n+1)!.
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 3 k + 1] a[k], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[_] = 0; Do[A[x_] = 1 + x A[x^3/(1 - x)^3]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 07 2022
STATUS
approved