A127680 -id:A127680 - cp's OEIS frontend

A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^2)).

1, 2, 4, 10, 24, 60, 148, 370, 920, 2296, 5720, 14268, 35568, 88700, 221156, 551482, 1375096, 3428888, 8549944, 21319624, 53160896, 132558360, 330537528, 824204780, 2055176304, 5124638944, 12778424976, 31863351980, 79452130896, 198116051644, 494007751668

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

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

Cf. A000621, A127680.

Cf. A218032, A319436.

nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x - x A[x^2]) + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 2] + Sum[a[Floor[k/2]] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 30}]

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)\2, v[j+1]*v[i-2*j])); v; \\ Seiichi Manyama, Nov 26 2023

G.f.: 1 / (1 - x - x / (1 - x^2 - x^2 / (1 - x^4 - x^4 / (1 - x^8 - x^8 / (1 - ...))))).
a(0) = 1, a(1) = 2; a(n) = a(n-2) + Sum_{k=0..n-1} a(floor(k/2)) * a(n-k-1).
a(n) ~ c * d^n, where d = 2.4935271724548067876965033643037290636931200352851874903211458249308... and c = 0.6156170089558875346518987360369130661426312977478830077668203229773... - Vaclav Kotesovec, Nov 16 2021
a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-1-2*k). - Seiichi Manyama, Nov 26 2023

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).

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).

A367655 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x) * A(x^2)).

Original entry on oeis.org

1, 2, 5, 14, 39, 111, 314, 894, 2539, 7224, 20536, 58413, 166102, 472410, 1343448, 3820748, 10865805, 30901790, 87882171, 249931270, 710786078, 2021427153, 5748794540, 16349171957, 46495891170, 132231060820, 376055838670, 1069476434880, 3041515866674

Offset: 0

Seiichi Manyama, Nov 26 2023

nonn

Cf. A367656, A367657, A367658.

Cf. A127680.

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\2+1]*v[i-j])); v;

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

A367747 E.g.f. satisfies A(x) = exp(x * (1 + x) * A(x^2)).

Original entry on oeis.org

1, 1, 3, 13, 73, 561, 4771, 49813, 562353, 7340833, 102829411, 1627648221, 27294311353, 502042022353, 9759264753603, 205434011254501, 4544894700204001, 107346788357502273, 2657668122191037763, 69701762677026498733, 1909106308252976007081

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A000272, A367748, A367749, A367751.

Cf. A127680, A138292.

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\2+1]*v[i-j]/((j\2)!*(i-1-j)!))); v;

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

A127681 a(0) = 1. a(n+1) = sum{k=0 to n} a(n-k)*a(ceiling(k/2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 42, 90, 198, 428, 936, 2030, 4430, 9626, 20978, 45622, 99367, 216197, 470736, 1024420, 2230183, 4853881, 10566170, 22997974, 50061240, 108964596, 237186018, 516272178, 1123772192, 2446081048, 5324371354, 11589437278

Offset: 0

Leroy Quet, Jan 23 2007

Robert Israel, Table of n, a(n) for n = 0..2958

Cf. A127680.

f:= proc(n) option remember;
    add(procname(n-1-k)*procname(ceil(k/2)),k=0..n-1)
end proc:
f(0):= 1:
map(f, [$0..40]); # Robert Israel, Feb 16 2018

f[l_List] := Block[{n = Length[l] - 1},Append[l, Sum[l[[n - k + 1]]*l[[Ceiling[k/2] + 1]], {k, 0, n}]]];Nest[f, {1}, 32] (* Ray Chandler, Feb 13 2007 *)

a(n) ~ c * d^n, where d = 2.17668434612191638687360948440303534082431658053308188275404767951385648... and c = 0.39120452795484998747876543545867360129596245925827624710922741574667... - Vaclav Kotesovec, Nov 16 2021

Extended by Ray Chandler, Feb 13 2007

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A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^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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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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A367655 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * (1 + x) * A(x^2)).

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A367747 E.g.f. satisfies A(x) = exp(x * (1 + x) * A(x^2)).

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A127681 a(0) = 1. a(n+1) = sum{k=0 to n} a(n-k)*a(ceiling(k/2)).

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