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 1-10 of 12 results. Next

A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 2, 15, 208, 4285, 117936, 4075099, 169736960, 8282604537, 463604723200, 29287449579751, 2061571190059008, 160023548976361525, 13580237335641417728, 1250935473495646861875, 124307671411309327876096, 13255531892787507819759601, 1509841440567809574906101760
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377831, A380664.
Cf. A052873, A380665.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x)))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n,n-k)/k!.

A377832 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x) ).

Original entry on oeis.org

1, 3, 29, 508, 13137, 452616, 19549021, 1016932512, 61940154177, 4325943203200, 340900244374461, 29927648769380352, 2896829645184711121, 306522175683831195648, 35201889560564096132925, 4360880891670519541927936, 579686447990401730151243009, 82304944815106131595482267648
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377831, A377833.
Cf. A377742, A377810.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(3*n-k+1, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.
a(n) ~ (1 + sqrt(3))^(4*n + 5/2) * n^(n-1) / (3^(1/4) * 2^(3*n + 5/2) * exp((sqrt(3) - 1)*n - 2 + sqrt(3))). - Vaclav Kotesovec, Nov 09 2024

A377859 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(x) ).

Original entry on oeis.org

1, 0, 1, 2, 21, 144, 1765, 21552, 340137, 5845760, 116495721, 2550320640, 62023290109, 1642735460352, 47321500546125, 1469008742856704, 48962556079079505, 1742660440701861888, 65993849612007279697, 2648999558505185280000, 112360563741545020804581
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377860, A377861.
Cf. A377831.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-1)^k*(n+1)^(k-1)*binomial(2*n-k, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(-x * A(x))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (-1)^k * (n+1)^(k-1) * binomial(2*n-k,n-k)/k!.
a(n) ~ phi^(3*n + 3/2) * n^(n-1) / (5^(1/4) * exp(phi*n + 1/phi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 10 2024

A377833 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(-x) ).

Original entry on oeis.org

1, 4, 51, 1174, 39833, 1799136, 101821723, 6938396368, 553482404721, 50619262481920, 5223014483031491, 600332651141435136, 76075005337204547209, 10538051760153093320704, 1584264031801742560408875, 256912816791069951740348416, 44703731640012047610981808097
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377831, A377832.
Cf. A377743, A377811.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(4*n-k+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x))^3.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(4*n-k+2,n-k)/k!.

A380723 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2) / (1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 21, 436, 13785, 589206, 31825381, 2080523880, 159761186577, 14097898530730, 1405926737063541, 156379679761925148, 19195200442017128425, 2577494115099820986174, 375845854490491567916805, 59145488004443221188738256, 9990898494797767848442559649, 1803160967691789114062089511250
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Cf. A377831, A380724.
Cf. A371318, A377888.
Cf. A360601, A380718.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(3*n-k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(3*n-k,n-k)/k!.

A377888 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 17, 289, 7541, 267041, 11974645, 650666731, 41560476809, 3052145052433, 253400719220801, 23470964805942083, 2399562226994185885, 268404500411311273465, 32606551238103342068717, 4275233840499570086190331, 601753408713140793660643985, 90500525005651471292191270433
Offset: 0

Views

Author

Seiichi Manyama, Nov 10 2024

Keywords

Crossrefs

Cf. A352410, A377831, A377889.
Cf. A371318.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n-k+1)^(k-1)*binomial(3*n-2*k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n-k+1)^(k-1) * binomial(3*n-2*k,n-k)/k!.

A380664 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)^2) ).

Original entry on oeis.org

1, 2, 17, 268, 6277, 196416, 7716109, 365398496, 20271580137, 1290027358720, 92653747607401, 7414981595716608, 654373744057368493, 63136350047908917248, 6612064512998173129125, 747016321343021395603456, 90564758322246657646854481, 11727981253987656671672008704
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377831, A380663.
Cf. A361598.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n+k, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x))^2)/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n+k,n-k)/k!.

A380724 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^3) / (1 - x*A(x)^3).

Original entry on oeis.org

1, 2, 29, 862, 39461, 2454296, 193406953, 18475039808, 2075062993865, 268013104242688, 39139481641977461, 6377306725457207552, 1147019426037344539501, 225728971809041691392000, 48248339461852786811399489, 11131014193619108036340637696, 2756799306857952163745291500433
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Cf. A377831, A380723.
Cf. A373324, A377889.
Cf. A360609.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(4*n-k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(4*n-k,n-k)/k!.

A382086 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 52, 845, 18816, 533617, 18404800, 748039833, 35016198400, 1855389108221, 109781344134144, 7174844881882405, 513331696318615552, 39905830821183755625, 3349445733955326754816, 301886246619209909215793, 29080090017105458412257280, 2981488457660004727761477493
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382087, A382088, A382089.
Cf. A000108, A377831, A382036.

Programs

  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(n+k-1, k)/(n-k-1)!));

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x) * exp(-x) ) ).
a(n) ~ phi^(3*n - 3/2) * n^(n-1) / (5^(1/4) * exp((n - 1/phi)/phi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 15 2025

A377889 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^3).

Original entry on oeis.org

1, 2, 21, 472, 16581, 795736, 48509641, 3589729760, 312603962985, 31321633489408, 3549706188092541, 448973808123051520, 62697159481460439469, 9581292408000225087488, 1590488540940006100524657, 284993765391981838318575616, 54826610288277007690469896017
Offset: 0

Views

Author

Seiichi Manyama, Nov 10 2024

Keywords

Crossrefs

Cf. A352410, A377831, A377888.
Cf. A373324.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (3*n-2*k+1)^(k-1)*binomial(4*n-3*k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n-2*k+1)^(k-1) * binomial(4*n-3*k,n-k)/k!.
Showing 1-10 of 12 results. Next

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