A147869 -id:A147869 - cp's OEIS frontend

A147953 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(n) = A147952(n).

1, 1, 1, 3, 4, 7, 9, 14, 22, 32, 43, 61, 89, 118, 167, 235, 312, 417, 572, 748, 1006, 1326, 1744, 2283, 2982, 3878, 5048, 6518, 8355, 10786, 13727, 17436, 22173, 28250, 35561, 45008, 56651, 70818, 88992, 111280, 138431, 172284, 214019, 265166, 328127

Offset: 0

Roger L. Bagula, Nov 17 2008

nonn

Cf. A147654, A147655, A147665, A147869, A147871, A147880, A147952.

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] = f[f[n - 2]] + If[Mod[n, 3] == 0,f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m] x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x],45]
(* Program edited and corrected by Petros Hadjicostas, Apr 12 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(1) = f(2) = 1, and for m >= 3, f(m) = f(f(m-2)) + r(m), where r(m) = f(f(floor(m/3)) when m == 0 or 1 (mod 3) and = f(m - f(floor(m/3))) when m == 2 (mod 3).

Various sections edited by Petros Hadjicostas, Apr 12 2020

A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 21, 29, 49, 73, 105, 162, 236, 338, 502, 706, 984, 1441, 1998, 2800, 3934, 5472, 7407, 10210, 14053, 19066, 25986, 35134, 47010, 63739, 85008, 112610, 150861, 200133, 264838, 349587, 459970, 602763, 792220, 1034136, 1345530

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

Cf. A004001, A005185, A147559, A147654, A147655, A147665, A147869, A147871.

f[n_Integer?Positive] := f[n] = f[n - f[n - 1]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1; (* A005185 *)
nmax = 41; CoefficientList[Series[Product[(1 + f[k] * x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Georg Fischer, Dec 10 2020 *)

\\ here B(n) is A005185 as vector.
B(n)={my(A=vector(n, k, 1)); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A}
seq(n)=my(v=B(n)); {Vec(prod(k=1, #v, 1 + x^k*v[k] + O(x*x^n)))} \\ Andrew Howroyd, Dec 10 2020

Definition corrected by Georg Fischer, Dec 10 2020

A147982 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(k) = A147952(A004001(k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 8, 11, 16, 20, 28, 38, 50, 69, 92, 120, 154, 203, 261, 338, 437, 559, 710, 907, 1146, 1444, 1829, 2291, 2863, 3593, 4457, 5539, 6882, 8503, 10501, 12931, 15861, 19466, 23854, 29125, 35520, 43279, 52557, 63735, 77358, 93472, 112885

Offset: 0

Roger L. Bagula, Nov 18 2008

nonn

Cf. A004001, A147869, A147952.

(* A004001 *) g[0] = 0; g[1] = 1; g[2] = 1; g[n_] := g[n] = g[g[n - 1]] + g[n - g[n - 1]];
(*A147952*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 2]] + If[Mod[n, 3] == 0, f[f[n/3]], If[Mod[n, 3] ==1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[g[m]]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified and corrected by Petros Hadjicostas, Apr 11 2020 using code from A147869 *)

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(k) = A147952(A004001(k)).

Various sections edited by Petros Hadjicostas, Apr 11 2020

A152006 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Original entry on oeis.org

1, 1, 2, 5, 8, 18, 34, 63, 102, 203, 336, 589, 999, 1675, 2799, 4768, 7561, 12224, 20513, 31724, 51621, 81976, 128560, 199192, 312536, 482806, 744847, 1147952, 1755931, 2649474, 4051413, 6069450, 9105323, 13747364, 20335077, 30508629, 45198631

Offset: 0

Roger L. Bagula, Nov 19 2008

nonn

Cf. A147557, A147654, A147655, A147665, A147869, A147871, A147880, A147953.

f[n_] = If[n < 2, n, Prime[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 37], x],37]
(* Program edited and corrected by Petros Hadjicostas, Apr 12 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Various sections edited by Petros Hadjicostas, Apr 12 2020

cp's OEIS Frontend

A147953 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(n) = A147952(n).

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A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

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A147982 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(k) = A147952(A004001(k)).

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A152006 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

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