A367653 -id:A367653 - cp's OEIS frontend

A127680 a(0) = 1; a(n+1) = Sum_{k=0..n} a(n-k)*a(floor(k/2)).

1, 1, 2, 4, 8, 17, 35, 74, 154, 324, 677, 1422, 2977, 6246, 13086, 27444, 57518, 120600, 252794, 529994, 1111013, 2329187, 4882755, 10236280, 21458943, 44986461, 94308415, 197707134, 414469000, 868886834, 1821517772, 3818600772

Offset: 0

Leroy Quet, Jan 23 2007

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A367652, A367653, A367654.

Cf. A127681.

f[l_List] := Block[{n = Length[l] - 1},Append[l, Sum[l[[n - k + 1]]*l[[Floor[k/2] + 1]], {k, 0, n}]]];Nest[f, {1}, 33] (* Ray Chandler, Feb 13 2007 *)

G.f. A(x) satisfies: A(x) = 1 / (1 - x * (1 + x) * A(x^2)). - Ilya Gutkovskiy, Nov 15 2021

a(n) ~ c * d^n, where d = 2.096382783759695271747034891835844892559952962948180418542044889824924... and c = 0.413348184087944400305975399220165744000861336139702047444087822224828... - Vaclav Kotesovec, Nov 16 2021

Extended by Ray Chandler, Feb 13 2007

A367654 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1025, 2051, 4104, 8212, 16432, 32882, 65798, 131664, 263464, 527200, 1054948, 2110989, 4224165, 8452706, 16914168, 33845864, 67726796, 135523764, 271187944, 542656864, 1085875984, 2172877052, 4348005437, 8700515871

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A127680, A367652, A367653.

Cf. A367661.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, v[j\5+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} a(floor(k/5)) * a(n-1-k).

A367652 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2) * A(x^3)).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 65, 131, 264, 534, 1078, 2176, 4396, 8877, 17925, 36202, 73108, 147636, 298152, 602108, 1215933, 2455552, 4958915, 10014374, 20223760, 40841302, 82477816, 166561622, 336366426, 679282324, 1371791274, 2770293218, 5594527784, 11297988864

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Cf. A127680, A367653, A367654.

Cf. A367659.

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

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

A367657 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x + x^2 + x^3) * A(x^4)).

Original entry on oeis.org

1, 2, 5, 13, 34, 90, 238, 629, 1662, 4394, 11616, 30707, 81173, 214584, 567259, 1499563, 3964128, 10479273, 27702219, 73231500, 193589270, 511758023, 1352844978, 3576279113, 9453982143, 24991835308, 66066533905, 174648514118, 461687660561, 1220482733670

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Cf. A367655, A367656, A367658.

Cf. A367653, A367660.

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

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

A367749 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3) * A(x^4)).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4777993, 62569891, 893927541, 13827333433, 234241234813, 4212828738483, 80727388033321, 1641227208417121, 35581993575319953, 810641581182744643, 19416795485684156893, 487647253209539939241

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A367747, A367748, A367751.

Cf. A367653, A367720.

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

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

A367794 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 346, 478, 660, 911, 1259, 1740, 2404, 3320, 4586, 6336, 8754, 12093, 16705, 23077, 31881, 44043, 60844, 84053, 116116, 160410, 221602, 306136, 422916, 584242, 807110, 1114996

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000108, A000621, A367637, A367800.

Cf. A367653, A367692, A367694, A367720.

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

from functools import lru_cache
@lru_cache(maxsize=None)
def A367794(n): return sum(A367794(k)*A367794(n-1-(k<<2)) for k in range(n+3>>2)) if n else 1 # Chai Wah Wu, Nov 30 2023

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

cp's OEIS Frontend

A127680 a(0) = 1; a(n+1) = Sum_{k=0..n} a(n-k)*a(floor(k/2)).

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A367654 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

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A367652 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2) * A(x^3)).

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A367657 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x + x^2 + x^3) * A(x^4)).

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A367749 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3) * A(x^4)).

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A367794 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^4)).

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