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 1-4 of 4 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

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

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
Showing 1-4 of 4 results.

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