A367654 -id:A367654 - 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

A367653 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2 + x^3) * A(x^4)).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 257, 515, 1032, 2068, 4146, 8310, 16656, 33384, 66916, 134125, 268837, 538850, 1080064, 2164860, 4339204, 8697416, 17432944, 34942268, 70037629, 140382111, 281379296, 563991416, 1130453878, 2265860666, 4541648896, 9103196384

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Cf. A127680, A367652, A367654.

Cf. A367660.

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\4+1]*v[i-j])); v;

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

A367658 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

Original entry on oeis.org

1, 2, 5, 13, 34, 89, 234, 615, 1616, 4246, 11156, 29314, 77026, 202394, 531811, 1397387, 3671781, 9647988, 25351094, 66612640, 175031647, 459913889, 1208471657, 3175385173, 8343655339, 21923823599, 57607130438, 151368736483, 397737124030, 1045095727865

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Cf. A367655, A367656, A367657.

Cf. A367654, A367661.

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\5+1]*v[i-j])); v;

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

A367751 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091, 844031541, 12949163833, 213873687613, 3782022682803, 71267635330921, 1439160383457121, 30612704101371153, 686728250047551043, 16198763975779425373, 400727742254252310441

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A367747, A367748, A367749.

Cf. A367654.

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\5+1]*v[i-j]/((j\5)!*(i-1-j)!))); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/5)) * a(n-1-k) / (floor(k/5)! * (n-1-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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A367653 G.f. A(x) satisfies A(x) = 1 / (1 - x * (1 + x + x^2 + x^3) * A(x^4)).

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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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A367658 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

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A367751 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3 + x^4) * A(x^5)).

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