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.

A376036 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) / (1 - A(x)).

Original entry on oeis.org

0, 1, 9, 190, 6435, 301126, 18007161, 1311752590, 112703870439, 11158543451926, 1250964512674533, 156642117419304958, 21668625406445359227, 3281750147124057118966, 540094007004476783547825, 95975344500184607391266734, 18314947854834472094038237647
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376034, A376035.
Cf. A371329, A376037.

Programs

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

Formula

a(n) = Sum_{k=1..n} (3*n+2*k-2)!/(3*n+k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x)^3 * log(1 + x * (1 - x)) ).

A274266 Expansion of e.g.f. (1 + x)^3*log(1 + x).

Original entry on oeis.org

1, 5, 11, 6, -6, 12, -36, 144, -720, 4320, -30240, 241920, -2177280, 21772800, -239500800, 2874009600, -37362124800, 523069747200, -7846046208000, 125536739328000, -2134124568576000, 38414242234368000, -729870602452992000, 14597412049059840000
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

First four terms [1, 5, 11, 6] form row 3 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 5*x^2/2 + 11*x^3/3! + 6*x^4/4! - 6*x^5/5! + ....

Crossrefs

Cf. A000169, A105954, A274265, A274268, A274270.

Programs

  • Magma
    [1,5,11] cat [(-1)^n*6*Factorial(n-4): n in [4..25]]; // Vincenzo Librandi, Jun 20 2016
  • Mathematica
    CoefficientList[Series[(1+t)^3 * Log[1+t], {t, 1, 20}], t]*Range[1, 20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^n*6*(n - 4)! for n >= 4.
E.g.f.: A(x) = (1 + x)^3*log(1 + x).
Series reversion(A(x)) = exp(-1/3*T(-3*x)) - 1 = x - 5*x^2/2! + 8^2*x^3/3! - 11^3*x^4/4! + 14^4*x^5/5! - ... is the e.g.f. for a signed version of A274265, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
Sum_{n>=1} 1/a(n) = 71/55 + 1/(6*e). - Amiram Eldar, Feb 02 2023

A274267 a(n) = (4*n - 1)^(n-1).

Original entry on oeis.org

1, 7, 121, 3375, 130321, 6436343, 387420489, 27512614111, 2251875390625, 208728361158759, 21611482313284249, 2472159215084012303, 309629344375621415601, 42141982597572021484375, 6193386212891813387462761, 977480813971145474830595007, 164890958756244164895763202881
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

Compare with A052774.

Links

Crossrefs

Cf. A000169, A052774, A274265, A274266, A274268, A274269.

Programs

  • Magma
    [(4*n-1)^(n-1): n in [1..25]]; // Vincenzo Librandi, Jun 20 2016
  • Maple
    A274267 := n -> (4*n - 1)^(n-1):
seq(A274267(n), n = 1..20);
  • Mathematica
    Table[(4*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)
  • PARI
    for(n=1,30, print1((4*n-1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

Formula

E.g.f. A(x) = 1 - exp(-1/4*T(4*x)) = x + 7*x^2/2! + 11^2*x^3/3! + 15^3*x^4/4! + 19^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^4*log(1/(1 - x)) ). See A274268.
1 - A(x) = exp(-x/(1 - A(x))^4) = exp(-x/(exp(-4*x/(exp(-4*x/ ...))))).
1 - A(-x*exp(4*x)) = exp(x) = 1/(1 - A(x*exp(-4*x))).
1/(1 - A(x)) = Sum_{n >= 0} (4*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052774.

A274269 a(n) = (5*n - 1)^(n-1).

Original entry on oeis.org

1, 9, 196, 6859, 331776, 20511149, 1544804416, 137231006679, 14048223625216, 1628413597910449, 210832519264920576, 30155888444737842659, 4722366482869645213696, 803596764671634487466709, 147653612273582215982104576, 29134419507545592909032289199
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

Compare with A052782.

Crossrefs

Cf. A000169, A052782, A274265, A274267, A274270.

Programs

Formula

E.g.f. A(x) = 1 - exp(-1/5*T(5*x)) = x + 9*x^2/2! + 14^2*x^3/3! + 19^3*x^4/4! + 24^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^5*log(1/(1 - x)) ). See A274270.
1 - A(x) = exp(-x/(1 - A(x))^5) = exp(-x/(exp(-5*x/(exp(-5*x/ ...))))).
1 - A(-x*exp(5*x)) = exp(x) = 1/(1 - A(x*exp(-5*x))).
1/(1 - A(x)) = Sum_{n >= 0} (5*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052782.

A274270 Expansion of e.g.f. (1 + x)^5*log(1 + x).

Original entry on oeis.org

1, 9, 47, 154, 274, 120, -120, 240, -720, 2880, -14400, 86400, -604800, 4838400, -43545600, 435456000, -4790016000, 57480192000, -747242496000, 10461394944000, -156920924160000, 2510734786560000, -42682491371520000, 768284844687360000, -14597412049059840000
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

The first six terms [1, 9, 47, 154, 274, 120] form row 5 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 9*x^2/2 + 47*x^3/3! + 154*x^4/4! + 274*x^5/5! + 120*x^6/6! - 120*x^7/7! + ....

Crossrefs

Cf. A000169, A105954, A274265, A274266, A274267, A274268, A274269, A274270.

Programs

  • Magma
    [1,9,47,154,274] cat [(-1)^n*120*Factorial(n - 6): n in [6..25]]; // Vincenzo Librandi, Jun 20 2016
  • Mathematica
    CoefficientList[Series[(1+t)^5 * Log[1+t], {t,1,20}],t]*Range[1,20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^n*120*(n - 6)! for n >= 6.
E.g.f.: A(x) = (1 + x)^5*log(1 + x).
Series reversion(A(x)) = exp(-1/5*T(-5*x)) - 1 = x - 9*x^2/2! + 14^2*x^3/3! - 19^3*x^4/4! + 24^4*x^5/5! - ... is the e.g.f. for a signed version of A274269, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
Sum_{n>=1} 1/a(n) = 5098232/4462227 + 1/(120*e). - Amiram Eldar, Feb 02 2023

A274268 Expansion of e.g.f. (1 + x)^4*log(1 + x).

Original entry on oeis.org

1, 7, 26, 50, 24, -24, 48, -144, 576, -2880, 17280, -120960, 967680, -8709120, 87091200, -958003200, 11496038400, -149448499200, 2092278988800, -31384184832000, 502146957312000, -8536498274304000, 153656968937472000, -2919482409811968000, 58389648196239360000
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

First five terms [1, 7, 26, 50, 24] form row 4 of A105954 read as a triangular array.

Examples

			E.g.f.= x + 7*x^2/2 + 26*x^3/3! + 50*x^4/4! + 24*x^5/5! - 24*x^6/6! + ...

Crossrefs

Cf. A000169, A105954, A274265, A274266, A274267, A274269, A274270.

Programs

  • Magma
    [1,7,26,50] cat [(-1)^(n-1)*24*Factorial(n-5): n in [5..25]]; // Vincenzo Librandi, Jun 20 2016
  • Mathematica
    CoefficientList[Series[(1+t)^4 * Log[1+t], {t, 1, 20}], t]*Range[1, 20]! (* G. C. Greubel, Jun 19 2016 *)

Formula

a(n) = (-1)^(n-1)*24*(n - 5)! for n >= 5.
E.g.f.: A(x) = (1 + x)^4*log(1 + x).
Series reversion(A(x)) = exp(-1/4*T(-4*x)) - 1 = x - 7*x^2/2! + 11^2*x^3/3! - 15^3*x^4/4! + 19^4*x^5/5! - ... is the e.g.f. for a signed version of A274267, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
Sum_{n>=1} 1/a(n) = 2733/2275 + 1/(24*e). - Amiram Eldar, Feb 02 2023

A376034 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) * (1 - A(x))^2.

Original entry on oeis.org

0, 1, 3, 28, 429, 9136, 249315, 8300692, 326261649, 14786485336, 759129218367, 43543567874764, 2759873588979045, 191549117617410736, 14448371199973057659, 1176874833493589697604, 102951969888432809238585, 9626512744249673928398920
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376035, A376036.

Programs

  • PARI
    a(n) = sum(k=1, n, (3*n-k-2)!/(3*n-2*k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (3*n-k-2)!/(3*n-2*k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x)^3 * log(1 + x / (1 - x)^2) ).

A376035 E.g.f. satisfies A(x) = exp(x / (1 - A(x))^3) - 1.

Original entry on oeis.org

0, 1, 7, 118, 3205, 120466, 5790619, 339216046, 23443311049, 1867308836986, 168435092561671, 16971155810393302, 1889194092179682061, 230257485553145337106, 30496977601634473249363, 4361533380688447142658046, 669865656003334085318195089
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376034, A376036.
Cf. A052892, A371371.

Programs

  • Mathematica
    Table[Sum[(3*n+k-2)!/(3*n-1)! * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 10 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(serreverse((1-x)^3*log(1+x)))))
  • PARI
    a(n) = sum(k=1, n, (3*n+k-2)!/(3*n-1)!*stirling(n, k, 2));

Formula

E.g.f.: Series_Reversion( (1 - x)^3 * log(1+x) ).
a(n) = Sum_{k=1..n} (3*n+k-2)!/(3*n-1)! * Stirling2(n,k).
a(n) ~ 3^(4*n-2) * LambertW(2*exp(1/3)/3)^(3*n-1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * exp(n) * 2^(3*n-1) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n-1)). - Vaclav Kotesovec, Sep 10 2024
Showing 1-8 of 8 results.

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