cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A300661 Expansion of e.g.f. exp(-Sum_{k>=1} prime(k)*x^k/k!).

Original entry on oeis.org

1, -2, 1, 5, 4, -53, -177, 282, 5759, 20355, -83420, -1420133, -6245485, 29035652, 648899541, 4034393367, -10488623858, -464971765297, -4310935438663, -3489419105786, 446500913437911, 6423072226704027, 30987397708208720, -462727554963927783, -11862200720684515159
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

Examples

			E.g.f.: A(x) = 1 - 2*x/1! + x^2/2! + 5*x^3/3! + 4*x^4/4! - 53*x^5/5! - 177*x^6/6! + 282*x^7/7! + ...

Links

Crossrefs

Cf. A000040, A007446, A007447, A030017, A030018, A300632.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*
      ithprime(j)*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[-Sum[Prime[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[-Prime[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: exp(-Sum_{k>=1} A000040(k)*x^k/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

Views

Author

Ilya Gutkovskiy, Apr 18 2018

Keywords

Examples

			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 + ...

Crossrefs

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

Programs

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

A307364 Expansion of 1/(1 - Sum_{k>=1} prime(k)#*x^k), where prime(k)# is the product of first k primes (A002110).

Original entry on oeis.org

1, 2, 10, 62, 454, 4310, 49954, 746078, 13180750, 283749638, 7747573666, 234558524690, 8437098259486, 340293472077722, 14523592739559970, 676119676949381762, 35425760935764788014, 2070535245695282709950, 125884029549845876309674, 8379955313909510350628018
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

Comments

Invert transform of A002110.

Crossrefs

Cf. A002110, A030017, A051296.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - Sum[Product[Prime[j], {j, k}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Product[Prime[j], {j, k}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A002110(k)*a(n-k).

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

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

A346792 G.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 3, 1, -1, -4, -10, 3, 0, 9, 19, 9, 2, -44, -27, -40, -3, 95, 75, 156, -36, -181, -274, -349, 81, 205, 982, 832, 35, -596, -2587, -1803, -1259, 2118, 5876, 5365, 4922, -6811, -12175, -17181, -12932, 14144, 28575, 53548, 27663, -19181
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A008578, A030017, A030018, A300662, A346791.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + x + Sum[Prime[k - 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

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

A353156 a(0) = 1; a(n) = -Sum_{k=1..n} prime(k+1) * a(n-k).

Original entry on oeis.org

1, -3, 4, -4, 2, 6, -22, 46, -74, 86, -40, -120, 450, -958, 1506, -1694, 744, 2500, -9184, 19422, -30450, 34032, -14178, -52286, 188038, -394724, 615102, -681110, 268666, 1089974, -3847390, 8021030, -12426638, 13632728, -5063588, -22711916, 78708912, -162966020, 251005706
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 27 2022

Keywords

Crossrefs

Cf. A006005, A030017, A030018, A065091, A292744, A300662, A307898, A307899, A346792.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = -Sum[Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[1/(1 + Sum[Prime[k + 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1 / (1 + Sum_{k>=1} prime(k+1) * x^k).
Previous Showing 11-16 of 16 results.

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