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.

A355110 Expansion of e.g.f. 2 / (3 - 2*x - exp(2*x)).

Original entry on oeis.org

1, 2, 10, 76, 768, 9696, 146896, 2596448, 52449536, 1191944704, 30097334784, 835973778432, 25330620762112, 831497823494144, 29394162040580096, 1113330929935101952, 44979662118902366208, 1930798895281527717888, 87756941394038739828736, 4210241529540625311727616
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A004123, A006155, A007405, A122704, A343672, A355111, A355112, A355113, A355114.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[2/(3 - 2 x - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 2^(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) * 2^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(3))) * ((3 - LambertW(exp(3)))/2)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

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

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

A367837 Expansion of e.g.f. 1/(2 - x - exp(4*x)).

Original entry on oeis.org

1, 5, 66, 1294, 33752, 1100504, 43060176, 1965653232, 102548623744, 6018735869824, 392498702352128, 28155539333730560, 2203322337542003712, 186790304541786160128, 17053569926181643921408, 1668166923908523824576512, 174057374767036007615922176
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A006155, A367835, A367836.
Cf. A285064, A343674, A355112, A367840.

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, 4^j*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^k * binomial(n,k) * a(n-k).

A367853 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)/4).

Original entry on oeis.org

1, 2, 12, 128, 1952, 38464, 926336, 26323968, 861419520, 31882358784, 1316275003392, 59954841649152, 2985997926727680, 161401148097036288, 9408988894966579200, 588381964243109412864, 39285329204482179858432, 2789234068575581984784384
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367851, A367852.
Cf. A352071, A355112, A367847.

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, 4^(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} 4^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-6 of 6 results.

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