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 21-27 of 27 results.

A143506 Irregular triangle read by rows: first row is 1, and n-th row gives the coefficients of x^(n - 1)*R(n,x + 1/x)/(x + 1/x), where R(n,x) is the n-th row polynomial for A060187.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 3, 6, 1, 1, 23, 26, 47, 26, 23, 1, 1, 76, 234, 304, 467, 304, 234, 76, 1, 1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1, 1, 722, 10549, 27158, 52730, 78586, 84365, 78586, 52730, 27158, 10549, 722, 1, 1, 2179, 60664, 272797, 563029, 1132234
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Keywords

Comments

Row sums yield A080253.

Examples

			Triangle begins:
   1;
   1,   1,    1;
   1,   6,    3,    6,    1;
   1,  23,   26,   47,   26,   23,    1;
   1,  76,  234,  304,  467,  304,  234,   76,    1;
   1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

Links

Crossrefs

Cf. A008292, A060187.
Cf. A143505, A143507.

Programs

  • Mathematica
    Table[CoefficientList[FullSimplify[ExpandAll[2^n*(1 - x - 1/x)^(1 + n)*x^n*LerchPhi[x + 1/x, -n, 1/2]]], x], {n, 0, 10}]//Flatten

Formula

Row n is generated by the polynomial 2^n*(1 - x - 1/x)^(1 + n)*x^n*Phi(x + 1/x, -n, 1/2), where Phi is the Lerch transcendant.
E.g.f.: (1 - x + x^2)*exp((1 + x + x^2)*t)/((1 + x^2)*exp(2*t*x) - x*exp(2*(1 + x^2)*t)). - Franck Maminirina Ramaharo, Oct 25 2018

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 25 2018

A162315 Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 1, 6, 1, 3, 3, 9, 1, 1, 12, 6, 12, 1, 3, 5, 30, 10, 15, 1, 1, 18, 15, 60, 15, 18, 1, 3, 7, 63, 35, 105, 21, 21, 1, 1, 24, 28, 168, 70, 168, 28, 24, 1, 3, 9, 108, 84, 378, 126, 252, 36, 27, 1, 1, 30, 45, 360, 210, 756, 210, 360, 45, 30, 1
Offset: 0

Views

Author

Peter Bala, Jul 01 2009

Keywords

Comments

Row reversed version of A124846. For the signless version of the inverse array and its connection with sums of powers of odd integers see A162313.

Examples

			Triangle begins
=================================================
n\k|..0.....1.....2.....3.....4.....5.....6.....7
=================================================
0..|..1
1..|..3.....1
2..|..1.....6.....1
3..|..3.....3.....9.....1
4..|..1....12.....6....12.....1
5..|..3.....5....30....10....15.....1
6..|..1....18....15....60....15....18.....1
7..|..3.....7....63....35...105....21....21.....1
...

Crossrefs

A007318, A151821 (row sums), A080253, A124846, A162313 (unsigned matrix inverse).

Programs

  • Maple
    #A162315
T:=(n, k)->(2-(-1)^(n-k))*binomial(n,k):
for n from 0 to 10 do seq(T(n,k), k = 0..n) od;

Formula

TABLE ENTRIES
(1)... T(n,k) = (2 - (-1)^(n-k))*binomial(n,k).
GENERATING FUNCTION
(2)... exp(x*t)*(2*exp(t)-exp(-t)) = 1 + (3+x)*t + (1+6*x+x^2)*t^2/2!
+ ....
The e.g.f. can also be written as
(3)... exp(x*t)/G(-t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f.
for A080253.
MISCELLANEOUS
The row polynomials form an Appell sequence of polynomials.
Row sums = A151821.

Extensions

Row sums corrected by Peter Bala, Apr 01 2010

A308864 a(n) = Sum_{k>=0} (n*k + 1)^n/2^(k+1).

Original entry on oeis.org

1, 2, 17, 442, 22833, 1942026, 245246761, 43001877122, 9986424563009, 2965574161158490, 1095862246322273601, 493067173454342315346, 265360795458419332828657, 168311426029488910748596394, 124248479512164840358578103577, 105608722927065949313865618984226
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Crossrefs

Cf. A000629, A000670, A080253, A285067, A307066.

Programs

  • Mathematica
    Table[Sum[(n k + 1)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 15}]
Table[n! SeriesCoefficient[Exp[x]/(2 - Exp[n x]), {x, 0, n}], {n, 0, 15}]
Join[{1}, Table[Sum[Binomial[n, k] n^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 1, 15}]]

Formula

a(n) = n! * [x^n] exp(x)/(2 - exp(n*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000670(k).
a(n) ~ sqrt(Pi/2) * n^(2*n + 1/2) / (log(2)^(n+1) * exp(n)). - Vaclav Kotesovec, Jun 29 2019

A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

Original entry on oeis.org

1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

Crossrefs

Cf. A000670, A001861, A052555, A080253, A083355, A216794, A346417, A346432.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001861(k) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * A000670(k).
a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021

A368438 Expansion of e.g.f. exp(x) / (3 - 2*exp(2*x)).

Original entry on oeis.org

1, 5, 49, 725, 14305, 352805, 10441489, 360526325, 14226677185, 631571457605, 31152937452529, 1690317145051925, 100052040656540065, 6415726051515362405, 443047566486308303569, 32780692416679034537525, 2587108326603274903810945
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A080253, A337026, A368439.
Cf. A368442.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+2*sum(j=1, i, 2^j*binomial(i, j)*v[i-j+1])); v;

Formula

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

A368439 Expansion of e.g.f. exp(x) / (4 - 3*exp(2*x)).

Original entry on oeis.org

1, 7, 97, 2023, 56257, 1955527, 81570337, 3969606823, 220777431937, 13813838801287, 960354511044577, 73441487204670823, 6126887497334519617, 553733062241471461447, 53894653951229272883617, 5620230710788836168938023, 625160202763025501303191297
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A080253, A337026, A368438.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+3*sum(j=1, i, 2^j*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + 3 * Sum_{k=1..n} 2^k * binomial(n,k) * a(n-k).

A337027 a(n) = (1/2) * Sum_{k>=0} (2*k + n)^n / 2^k.

Original entry on oeis.org

1, 3, 24, 293, 4784, 97687, 2393472, 68405073, 2233928448, 82063263371, 3349249267712, 150353137462717, 7362889615257600, 390601858379350815, 22315011551291080704, 1365896953310909493929, 89179296762081886011392, 6186383336743041502051219
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

Crossrefs

Cf. A080253, A162314, A216794, A292916.

Programs

  • Mathematica
    Table[2^(n - 1) HurwitzLerchPhi[1/2, -n, n/2], {n, 0, 17}]
Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[2 x]), {x, 0, n}], {n, 0, 17}]

Formula

a(n) = n! * [x^n] exp(n*x) / (2 - exp(2*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A216794(n-k).
Previous Showing 21-27 of 27 results.

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