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.

Previous Showing 11-18 of 18 results.

A118793 Triangle where T(n,k) = -n!/(n-k)!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

Original entry on oeis.org

1, -1, 3, 1, -9, 19, -1, 18, -103, 207, 1, -30, 325, -1605, 3211, -1, 45, -785, 6930, -32191, 64383, 1, -63, 1610, -22050, 175861, -790629, 1581259, -1, 84, -2954, 57750, -693861, 5216778, -22974463, 45948927, 1, -108, 4998, -131922, 2213211, -24542910, 177555925, -770820885, 1541641771
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2006

Keywords

Comments

[0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A118794 with e.g.f.: 1-exp((-1+sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118795 with e.g.f.: -1+exp((1-sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is to be offset with leading zero.

Examples

			Triangle begins:
1;
-1, 3;
1,-9, 19;
-1, 18,-103, 207;
1,-30, 325,-1605, 3211;
-1, 45,-785, 6930,-32191, 64383;
1,-63, 1610,-22050, 175861,-790629, 1581259;
-1, 84,-2954, 57750,-693861, 5216778,-22974463, 45948927; ...
which is formed from the powers of F(x) = x/log(1-x-x^2):
F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
F(x)^2 = ( 1 - 3*x) + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
F(x)^3 = (-1 + 9/2*x - 19/2*x^2) + 15*x^3 - 5161/240*x^4 +...
F(x)^4 = ( 1 - 18/3*x + 103/6*x^2 - 207/6*x^3) + 42239/720*x^4 +...
F(x)^5 = (-1 + 30/4*x - 325/12*x^2 + 1605/24*x^3 - 3211/24*x^4) +...

Crossrefs

Cf. A052886 (diagonal), A118794 (row sums), A118795 (unsigned row sums); A118791 (variant).

Programs

  • PARI
    {T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!/(n-k)!*polcoeff(((x/log(1-x-x^2)))^(n+1),k,X)}

A345102 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 37, 589, 13276, 386059, 13741057, 578451514, 28109736811, 1548565036789, 95365652263102, 6492034471389889, 484086370908869821, 39238367740327468444, 3435176518078688461297, 323029539924876486293089, 32472511993953383052630556, 3475005417300807667690138399
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

Links

Crossrefs

Cf. A006677, A011781, A052886, A201354, A345103.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[7 - 6 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 3^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(7-6*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / sqrt(7 - 6 * exp(x)).

A118791 Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

Original entry on oeis.org

1, -1, 3, 2, -9, 19, -6, 36, -103, 207, 24, -180, 650, -1605, 3211, -120, 1080, -4710, 13860, -32191, 64383, 720, -7560, 38640, -132300, 351722, -790629, 1581259, -5040, 60480, -354480, 1386000, -4163166, 10433556, -22974463, 45948927, 40320, -544320, 3598560, -15830640, 53117064
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2006

Keywords

Comments

[0, diagonal] = A052886 with e.g.f.: (1-sqrt(5-4*exp(x)))/2. [0, row sums] = A117271 with e.g.f.: log((3-sqrt(5-4*exp(x)))/2). [0, unsigned row sums] = A118792 with e.g.f.: -log((1+sqrt(5-4*exp(x)))/2). Here [0, sequence] indicates that the sequence is offset with a leading zero.

Examples

			Triangle begins:
1;
-1, 3;
2,-9, 19;
-6, 36,-103, 207;
24,-180, 650,-1605, 3211;
-120, 1080,-4710, 13860,-32191, 64383;
720,-7560, 38640,-132300, 351722,-790629, 1581259;
-5040, 60480,-354480, 1386000,-4163166, 10433556,-22974463, 45948927;
which is formed from the powers of F(x) = x/log(1-x-x^2):
F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
F(x)^2 = ( 1 - 3*x)/1! + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
F(x)^3 = (-2 + 9*x - 19*x^2)/2! + 15*x^3 - 5161/240*x^4 +...
F(x)^4 = ( 6 - 36*x + 103*x^2 - 207*x^3)/3! + 42239/720*x^4 +...
F(x)^5 = (-24 + 180*x - 650*x^2 + 1605*x^3 - 3211*x^4)/4! +...

Crossrefs

Cf. A052886 (diagonal), A117271 (row sums), A118792 (unsigned row sums); A118793 (variant).

Programs

  • PARI
    {T(n,k)=local(x=X+X^2*O(X^(k+2)));-n!*polcoeff(((x/log(1-x-x^2)))^(n+1),k,X)}

A335441 a(n) = 1 + Sum_{k=1..n-1} binomial(n-2,k-1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 2, 4, 11, 40, 176, 907, 5360, 35668, 263789, 2146390, 19054040, 183248581, 1897952690, 21061861828, 249309196559, 3135518918800, 41754612283244, 586922460056851, 8684272948653068, 134919751191875572, 2195942678525060093, 37365571515146318650
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 10 2020

Keywords

Links

Crossrefs

Cf. A000111, A006677, A052886, A054687, A080635, A332776.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k - 1] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[Integrate[Exp[x] + A[x] D[A[x], x], x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x) * A'(x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: (BesselY(0, sqrt(2))*(BesselJ(1, sqrt(2)*exp(x/2)) - sqrt(2)*exp(x/2)*BesselJ(0, sqrt(2)*exp(x/2))) + BesselJ(0, sqrt(2))*(sqrt(2)*exp(x/2)*BesselY(0, sqrt(2)*exp(x/2)) - BesselY(1, sqrt(2)*exp(x/2)))) / (BesselJ(1, sqrt(2)*exp(x/2))*BesselY(0, sqrt(2)) - BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(x/2))).
a(n) ~ 2 * n! / r^(n+1), where r = 1.35169030867903432729790416904526340210784862703704233748118252928787... is the smallest real root of the equation BesselY(0, sqrt(2))*BesselJ(1, sqrt(2)*exp(r/2)) = BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(r/2)). (End)

A345103 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 5, 61, 1277, 37741, 1437725, 67013101, 3693540317, 234974905261, 16945434018845, 1366008048556141, 121721015465713757, 11880107754103150381, 1260413749895624939165, 144427420001275864755181, 17776090894283922227621597, 2338833689096321086977341101, 327585830473259220341296486685
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

Links

Crossrefs

Cf. A006677, A052886, A144828, A201365, A345102.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(9-8*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / sqrt(9 - 8 * exp(x)).

A345104 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 3, 13, 89, 825, 9601, 134185, 2188353, 40788745, 855303265, 19927758377, 510728051073, 14279388168137, 432505475357729, 14107767947949289, 493046896702987841, 18380057918926012809, 728005164671113691105, 30531323352522247757225, 1351567976217998536472833
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

Crossrefs

Cf. A000165, A052886, A054687, A345105.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 2 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 1; Do[A[x] = Normal[Integrate[2 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

Formula

E.g.f. A(x) satisfies: A'(x) = 2 * A(x)^2 + exp(x).

A371316 E.g.f. satisfies A(x) = (exp(x) - 1)/(1 - A(x))^2.

Original entry on oeis.org

0, 1, 5, 55, 1001, 25471, 832265, 33209695, 1565233241, 85089724831, 5241027586025, 360724089079135, 27436914192242681, 2285358551395272991, 206893372546088226185, 20226992715373747441375, 2123855112711652849031321, 238375283773978224211297951
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Links

Crossrefs

Cf. A052886, A371317.

Programs

  • Mathematica
    Table[Sum[(3*k-2)!/(2*k-1)! * StirlingS2[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 19 2024 *)
  • PARI
    a(n) = sum(k=1, n, (3*k-2)!/(2*k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (3*k-2)!/(2*k-1)! * Stirling2(n,k).
a(n) ~ sqrt(31) * n^(n-1) / (sqrt(2) * 3^(3/2) * log(31/27)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Mar 19 2024
E.g.f.: Series_Reversion( log(1 + x * (1 - x)^2) ). - Seiichi Manyama, Sep 08 2024

A371317 E.g.f. satisfies A(x) = (exp(x) - 1)/(1 - A(x))^3.

Original entry on oeis.org

0, 1, 7, 109, 2767, 97621, 4411927, 243200749, 15823364767, 1186906701061, 100841639699047, 9571637343623389, 1003836707927846767, 115277656980955696501, 14386616557957088368567, 1938794142849968447054029, 280599557966128999505000767
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Links

Crossrefs

Cf. A052886, A371316.

Programs

  • PARI
    a(n) = sum(k=1, n, (4*k-2)!/(3*k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (4*k-2)!/(3*k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( log(1 + x * (1 - x)^3) ). - Seiichi Manyama, Sep 08 2024
Previous Showing 11-18 of 18 results.

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

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