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.

A377554 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 30, 537, 14124, 493695, 21601458, 1137294039, 70064934600, 4947238170747, 394022075650590, 34951812094581723, 3417754921150904172, 365287875167708973831, 42368411854713294141834, 5300422308901745571018735, 711465905597330333014408848, 101995745742232833085109746803
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2024

Keywords

Links

Crossrefs

Cf. A161633, A377553.
Cf. A364986.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364986.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(3*n+3,k)/(n-k)!.

A382032 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x))^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 55, 937, 21741, 639841, 22839139, 958882289, 46304377849, 2528571710881, 154076164781991, 10364272238514217, 762867688235619877, 60989719558159065857, 5263030218009265964011, 487578723768665716788961, 48266847740986728218648433, 5084697384633390178057209793
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161630, A382033, A382034.
Cf. A000108, A377553, A382036.

Programs

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

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377553. F(x) = log(A(x))/x = C(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^2 ) ).

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 14, 150, 2264, 44370, 1073772, 30998954, 1041094448, 39909978594, 1720526113460, 82422717484602, 4345035540566184, 250012958308399442, 15594180423126432428, 1048169467357831893930, 75535629221800163853152, 5810132660615400890909634, 475146028302302130377698404
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377575, A377576.
Cf. A002999, A295238, A377553.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A295238.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+2,k)/( (k+1)*(n-k)! ).
a(n) ~ 2^(5/2) * sqrt(1 + LambertW(1/4)) * n^(n-1) / (LambertW(1/4)^n * exp(n)). - Vaclav Kotesovec, Nov 02 2024

A377630 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 52, 1212, 41512, 1889700, 107684664, 7384011796, 592485333472, 54488274328836, 5652345176418280, 653054114586249684, 83175314479016845584, 11578838832843098353732, 1749242011108507789948312, 285034599164755404426493140, 49833544890911336997795542464
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A161633, A377553, A377554.
Cf. A364989, A377633.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364989.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(4*n+4,k)/(n-k)!.

A381519 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sin(x))^2 ).

Original entry on oeis.org

1, 2, 10, 82, 936, 13642, 240656, 4952218, 115608704, 2992207250, 84070140672, 2507383885730, 77117178496000, 2329071118971482, 61202811821836288, 690380688651775978, -88097620429234470912, -11900508444760552311518, -1112180862634722333884416
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2025

Keywords

Links

Crossrefs

Cf. A136630, A381389, A381518.
Cf. A377553, A381442, A381449, A381521.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + sin(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381518.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A379861 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x)/(1 + x*exp(x))^2 ).

Original entry on oeis.org

1, 1, 5, 38, 441, 6714, 128245, 2943562, 79049201, 2432351618, 84408126621, 3261942050058, 138946757581225, 6468600047278498, 326782092756236741, 17805164917279808234, 1040857709162817298401, 64983981546315031200258, 4315627103007355018430509
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A377553, A379862.

Programs

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

Formula

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

A379862 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x*exp(x))^2 ).

Original entry on oeis.org

1, 3, 29, 502, 12761, 430986, 18217813, 926514058, 55133781809, 3760088111938, 289240874117981, 24780044801646762, 2340229465310736073, 241563626661550193794, 27059024800372108029221, 3269263894468329061597546, 423798837014001794141132897, 58674726188995774863597090690
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A377553, A379861.
Cf. A379456.

Programs

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

Formula

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

A381521 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cos(x))^2 ).

Original entry on oeis.org

1, 2, 10, 78, 792, 9250, 106080, 636286, -30646784, -2237508990, -112000654080, -5124930562642, -227068649702400, -9819508698442846, -406371251899045888, -15094508095346343330, -394372545425757634560, 7096803535075158290434, 2430273114806112504446976
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A185951, A381480, A381520.
Cf. A377553, A381442, A381449, A381519.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*I^(n-k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cos(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381520.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A185951(n,k), where i is the imaginary unit.
Showing 1-8 of 8 results.

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