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 1-8 of 8 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

A380665 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 3, 31, 586, 16401, 612336, 28678231, 1618268688, 106946168769, 8105456425600, 693228400344591, 66055574392722432, 6940237183385667409, 797165049089377683456, 99381018789002592800775, 13365207839280075801020416, 1928719845703457066672384769, 297293268794967068206087176192
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A052873, A380663.
Cf. A377832.

Programs

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

Formula

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

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!.

A380674 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)^2) ).

Original entry on oeis.org

1, 3, 25, 370, 8097, 237096, 8733601, 388380000, 20253654945, 1212334652800, 81937521020841, 6172429566120192, 512850795552978625, 46594245206418954240, 4595466275857015549425, 488993161791784338804736, 55839856392986843905585089, 6811561624203525171739852800
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377832, A380665, A380666, A380675.

Programs

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

A382087 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^2) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 106, 2525, 82536, 3436867, 174045376, 10385025849, 713599868800, 55498397386751, 4819444051348224, 462246012357060373, 48531686994029295616, 5536163290789601602875, 681824639839489261060096, 90168540044259473683829873, 12744019609725371553920876544
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382086, A382088, A382089.
Cf. A001764, A377832, A382037.

Programs

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

Formula

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

A377860 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(x) ).

Original entry on oeis.org

1, 1, 5, 44, 577, 10104, 222133, 5886880, 182775969, 6509571200, 261665344261, 11720054882304, 578878362625825, 31259890045425664, 1832295378792935925, 115862322601669627904, 7861907382202262095297, 569837358810005613281280, 43939338917141224534941829
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377859, A377861.
Cf. A377832.

Programs

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

A380675 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)) ).

Original entry on oeis.org

1, 3, 27, 436, 10377, 329016, 13079971, 626414496, 35132554449, 2259697340800, 164013549475371, 13263204195136512, 1182645846100592473, 115285805003164594176, 12197859187688440506675, 1392237638583170475298816, 170517388925776876433310369, 22307473046095249063001554944
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377832, A380665, A380666, A380674.

Programs

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

Formula

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

A380945 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-2*x) ).

Original entry on oeis.org

1, 4, 50, 1124, 37192, 1637232, 90278176, 5992556320, 465599728512, 41470892979200, 4167168740195584, 466428111222196224, 57556315795242096640, 7763511917730857967616, 1136484206117494859980800, 179453678311835212416585728, 30404317385796994658988752896
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2025

Keywords

Links

Crossrefs

Cf. A370054, A377832.
Cf. A380646, A380723.

Programs

  • PARI
    a(n, q=2, r=2, s=2, 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(2*x*A(x))/(1 - x*A(x))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380723.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(3*n-k+1,n-k)/k!.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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