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.

Showing 1-6 of 6 results.

A367835 Expansion of e.g.f. 1/(2 - x - exp(2*x)).

Original entry on oeis.org

1, 3, 22, 242, 3544, 64872, 1424976, 36517840, 1069533824, 35240047232, 1290137297152, 51955085596416, 2282489348834304, 108630445541684224, 5567741266098944000, 305752314499878569984, 17909736027185859100672, 1114647522476340562132992
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A006155, A367836, A367837.
Cf. A126390, A216794, A343672, A355110, A367838.

Programs

  • Maple
    A367835 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        n*procname(n-1)+add(2^k*binomial(n,k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367835(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^j*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^k * binomial(n,k) * a(n-k).

A355111 Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).

Original entry on oeis.org

1, 2, 11, 93, 1041, 14541, 243747, 4767183, 106556373, 2679469065, 74864397015, 2300883358995, 77144051804409, 2802027511061325, 109604157405491691, 4593512301562215783, 205348466229473678301, 9753645833118762303249, 490530576727430107027839, 26040317900991310393061499
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 3^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A355112 Expansion of e.g.f. 4 / (5 - 4*x - exp(4*x)).

Original entry on oeis.org

1, 2, 12, 112, 1376, 21056, 386688, 8286720, 202958848, 5592199168, 171203895296, 5765504860160, 211811563929600, 8429932686999552, 361312700788375552, 16592261047219388416, 812749365813312487424, 42299637489384965537792, 2330989060564353634271232
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003576, A006155, A094417, A326324, A343674, A355110, A355111, A355113, A355114.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[4/(5 - 4 x - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(5))) * ((5 - LambertW(exp(5)))/4)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A355113 Expansion of e.g.f. 5 / (6 - 5*x - exp(5*x)).

Original entry on oeis.org

1, 2, 13, 133, 1779, 29565, 589705, 13728695, 365295695, 10934634985, 363678872325, 13305294463275, 531030788556475, 22960273845453725, 1069101897816615425, 53336480697298243375, 2838300249311563302375, 160480124820425410172625, 9607441647405962075600125
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003577, A006155, A094418, A355110, A355111, A355112, A355114.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[5/(6 - 5 x - Exp[5 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 5^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 5^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(6))) * ((6 - LambertW(exp(6)))/5)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A355114 Expansion of e.g.f. 6 / (7 - 6*x - exp(6*x)).

Original entry on oeis.org

1, 2, 14, 156, 2256, 40416, 869040, 21817440, 626063616, 20210176512, 724888631808, 28599923045376, 1230970377166848, 57397448756994048, 2882187551571941376, 155065468075097960448, 8898907099302329647104, 542609247778976191610880, 35031706496702707368591360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003578, A006155, A094419, A355110, A355111, A355112, A355113.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[6/(7 - 6 x - Exp[6 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 6^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(7))) * ((7 - LambertW(exp(7)))/6)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A367851 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 10, 80, 872, 11984, 198416, 3840192, 85031040, 2119385856, 58714881792, 1789646610432, 59515302478848, 2144299161348096, 83204666280609792, 3459286210445942784, 153413140701637804032, 7228914528043587796992, 360670654712328998289408
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367852, A367853.
Cf. A227917, A355110, A367845.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-6 of 6 results.

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