A147655 -id:A147655 - cp's OEIS frontend

A304791 Expansion of Product_{k>=1} (1 - prime(k)*x^k).

1, -2, -3, 1, 3, 18, 0, 35, -27, -85, -91, -109, -366, 118, 942, -957, 2791, 2091, 4855, -1157, -6903, 3341, 3162, -37034, -46480, -89890, 581, 131275, -296935, 167543, 108671, 801491, 616017, 2441581, -307733, -1864550, 4495872, 1158228, -2589768, -767646, -21062537

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A145519.

Index entries for sequences related to partitions

Cf. A000040, A007441, A145519, A147655, A298159.

nmax = 40; CoefficientList[Series[Product[(1 - Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d Prime[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: Product_{k>=1} (1 - A000040(k)*x^k).

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

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

A371310 Expansion of e.g.f. Product_{k>=1} (1 + prime(k)*x^k/k!).

Original entry on oeis.org

1, 2, 3, 23, 47, 231, 2260, 6527, 35151, 224759, 3434124, 12476055, 79758206, 491191521, 4752819625, 105146082344, 393097093065, 2976053272527, 21569670506914, 188844207315245, 2277243901499454, 72603521472295945, 326137558352646889, 2491611720654851668

Offset: 0

Ilya Gutkovskiy, Mar 24 2024

nonn

"EFJ" (unordered, size, labeled) transform of primes.

C. G. Bower, Transforms (2)

Cf. A000040, A007837, A032299 A032300, A147655.

nmax = 23; CoefficientList[Series[Product[(1 + Prime[k] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A304791 Expansion of Product_{k>=1} (1 - prime(k)*x^k).

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

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Crossrefs

Programs

Mathematica

Formula

Extensions

A371310 Expansion of e.g.f. Product_{k>=1} (1 + prime(k)*x^k/k!).

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Mathematica