A351973 -id:A351973 - cp's OEIS frontend

A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

1, 2, 3, 6, 11, 21, 40, 78, 151, 294, 572, 1115, 2172, 4234, 8252, 16088, 31361, 61140, 119191, 232370, 453010, 883167, 1721768, 3356675, 6543988, 12757830, 24871992, 48489172, 94531974, 184294706, 359291464, 700456240, 1365573493, 2662252082, 5190190005

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Cf. A000621, A007317, A351973.

a[n_] := a[n] = 1 + Sum[a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x^2])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x^2))).

A367694 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * A(x^4))).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 8, 10, 13, 16, 20, 24, 29, 35, 43, 53, 66, 82, 102, 126, 155, 190, 233, 286, 352, 435, 537, 664, 819, 1011, 1244, 1532, 1884, 2322, 2860, 3528, 4349, 5366, 6614, 8154, 10044, 12377, 15247, 18791, 23156, 28546

Offset: 0

Seiichi Manyama, Nov 27 2023

nonn

Cf. A005043, A351973, A367693.

Cf. A367660, A367692.

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+sum(j=0, (i-1)\4, v[j+1]*v[i-4*j])); v;

a(n) = (-1)^n + Sum_{k=0..floor((n-1)/4)} a(k) * a(n-1-4*k).

A367693 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * A(x^3))).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 2, 2, 3, 3, 4, 5, 7, 9, 12, 15, 19, 24, 31, 40, 53, 68, 88, 113, 145, 186, 241, 311, 402, 519, 669, 861, 1110, 1431, 1846, 2382, 3073, 3962, 5109, 6586, 8492, 10952, 14125, 18216, 23493, 30294, 39063, 50373, 64959, 83769, 108030, 139314

Offset: 0

Seiichi Manyama, Nov 27 2023

nonn

Cf. A005043, A351973, A367694.

Cf. A367659, A367691.

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+sum(j=0, (i-1)\3, v[j+1]*v[i-3*j])); v;

a(n) = (-1)^n + Sum_{k=0..floor((n-1)/3)} a(k) * a(n-1-3*k).

A367716 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * (1 + x) * A(x^2))).

Original entry on oeis.org

1, 0, 2, 1, 4, 6, 13, 23, 44, 82, 154, 292, 547, 1036, 1943, 3672, 6900, 13022, 24498, 46194, 86958, 163892, 308624, 581532, 1095275, 2063534, 3886876, 7322523, 13793363, 25984580, 48948062, 92209073, 173699564, 327214934, 616397498, 1161163428, 2187371054

Offset: 0

Seiichi Manyama, Nov 28 2023

nonn

Cf. A005043, A367717, A367718.

Cf. A351973, A367713.

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+sum(j=0, i-1, v[j\2+1]*v[i-j])); v;

a(n) = (-1)^n + Sum_{k=0..n-1} a(floor(k/2)) * a(n-1-k).

cp's OEIS Frontend

A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

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A367694 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * A(x^4))).

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A367693 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * A(x^3))).

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PARI

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A367716 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - x * (1 + x) * A(x^2))).

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