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 1-5 of 5 results.

A377831 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x) ).

Original entry on oeis.org

1, 2, 13, 154, 2701, 63216, 1856569, 65711024, 2724349401, 129552751360, 6952877604421, 415770771875328, 27416031835737637, 1976460653044957184, 154658036515292528625, 13055394531339601033216, 1182611605875201470044081, 114426900236922150187892736
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A001622, A377832, A377833.

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)).
a(n) = n! * Sum_{k=0..n} (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((n+1)/phi - 1)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 09 2024

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

A382088 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 178, 5549, 237456, 12945037, 858203872, 67035559257, 6029839290880, 613862192499281, 69777500840918784, 8760124051527691141, 1203852634738613966848, 179746834136205848167125, 28975042890917781500747776, 5015346425440407318539964593, 927775677566572703009955053568
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382086, A382087, A382089.
Cf. A002293, A377833, A382038.

Programs

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

Formula

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

A377861 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(x) ).

Original entry on oeis.org

1, 2, 15, 206, 4193, 113904, 3882511, 159475280, 7672503681, 423360926720, 26362968645071, 1829066086810368, 139929538526047585, 11703312997355442176, 1062423600515479191375, 104042389901715413633024, 10933256593926589800851969, 1227201235266954603172331520
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377859, A377860.
Cf. A377833.

Programs

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

A380946 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(-3*x) ).

Original entry on oeis.org

1, 6, 111, 3678, 179073, 11588688, 938905551, 91542271824, 10444685410881, 1365936450693120, 201503447217869679, 33108736185915906816, 5997057218957213126721, 1187319940110958086623232, 255104922613608981003351375, 59120580081196768991316314112
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2025

Keywords

Links

Crossrefs

Cf. A370057, A377833.
Cf. A380647, A380724.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x))/(1 - x*A(x))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380724.
a(n) = 3 * n! * Sum_{k=0..n} (3*n+3)^(k-1) * binomial(4*n-k+2,n-k)/k!.
Showing 1-5 of 5 results.

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