A302194 -id:A302194 - cp's OEIS frontend

A307770 Expansion of e.g.f. 1/(1 - Sum_{k>=1} prime(k)*x^k/k!).

1, 2, 11, 89, 957, 12871, 207717, 3910931, 84155053, 2037195551, 54795228241, 1621233039941, 52328310410427, 1829742961027269, 68901415049874055, 2779901582389463177, 119635322278784511015, 5470390958849723994819, 264850557367286330886261, 13535194864326763053170325

Offset: 0

Ilya Gutkovskiy, Apr 27 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000040, A007446, A030017, A302191, A302192, A302194.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n, j)*ithprime(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 24 2021

nmax = 19; CoefficientList[Series[1/(1 - Sum[Prime[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

A346791 E.g.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, -1, 0, 3, -5, -17, 103, 57, -2707, 6785, 84135, -659369, -2129683, 55537445, -103722105, -4630217025, 37357780827, 334163569535, -7214177094045, -2126819153101, 1233139349668817, -8794491537166765, -184459444459530193, 3483053621920936363, 15570880115951580635

Offset: 0

Ilya Gutkovskiy, Aug 04 2021

sign

Cf. A000040, A008578, A030018, A300632, A300662, A302194, A346430.

nmax = 24; CoefficientList[Series[1/(1 + x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * A008578(k) * a(n-k).

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)x^k) = Sum_{n>=1} a(n)x^n/n.

Original entry on oeis.org

2, 2, 5, 2, 12, -13, 16, -30, 41, -18, 46, -73, 132, -278, 315, -318, 580, -805, 1218, -1998, 2665, -3958, 5936, -7761, 11612, -17678, 25313, -38134, 54754, -76833, 114392, -166334, 240685, -356454, 515996, -748441, 1095572, -1581482, 2303163, -3375550, 4903684, -7149365, 10417010, -15111622

Offset: 1

Ilya Gutkovskiy, Apr 18 2018

sign

			L.g.f.: L(x) = 2*x + 2*x^2/2 + 5*x^3/3 + 2*x^4/4 + 12*x^5/5 - 13*x^6/6 + 16*x^7/7 - 30*x^8/8 + 41*x^9/9 - 18*x^10/10 + ...
exp(L(x)) = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + 23*x^9 + 29*x^10 + ... + A000040(n)*x^n + ...

Cf. A000040, A007446, A007447, A008578, A030017, A030018, A302194.

nmax = 44; Rest[CoefficientList[Series[Log[1 + Sum[Prime[k] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

A346430 E.g.f.: 1 / (1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, 1, 4, 21, 149, 1317, 13985, 173207, 2451807, 39043963, 690844441, 13446183857, 285500221447, 6567135007015, 162678487750465, 4317650962178897, 122234460353464081, 3676789159574231397, 117102826395968235853, 3936834192059910096205, 139316727760914366716635

Offset: 0

Ilya Gutkovskiy, Aug 04 2021

nonn

Cf. A000040, A008578, A030017, A300632, A300662, A302194, A346791.

nmax = 20; CoefficientList[Series[1/(1 - x - Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A008578(k) * a(n-k).

cp's OEIS Frontend

A307770 Expansion of e.g.f. 1/(1 - Sum_{k>=1} prime(k)*x^k/k!).

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A346791 E.g.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k / k!).

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A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)x^k) = Sum_{n>=1} a(n)x^n/n.

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A346430 E.g.f.: 1 / (1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

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cp's OEIS Frontend

A307770 Expansion of e.g.f. 1/(1 - Sum_{k>=1} prime(k)*x^k/k!).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A346791 E.g.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k / k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)*x^k) = Sum_{n>=1} a(n)*x^n/n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A346430 E.g.f.: 1 / (1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)x^k) = Sum_{n>=1} a(n)x^n/n.