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 1-5 of 5 results.

A097181 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).

Original entry on oeis.org

1, 1, 14, 1, 21, 210, 1, 28, 378, 3220, 1, 35, 595, 6475, 49910, 1, 42, 861, 11396, 108402, 778596, 1, 49, 1176, 18326, 207074, 1791930, 12198004, 1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920, 1, 63, 1953, 39585, 587727, 6783147
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Comments

Row sums form A097185. Diagonal is A097183.
Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097184, where the g.f.s satisfy: A097182(x*A097184(x)) = A097184(x).

Examples

			Row polynomials evaluated at y=1/2 equals powers of 8:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
where A097182(y)^(n+1) has the same initial terms as the n-th row:
A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
A097182(y)^2 = 1 + 14y +...
A097182(y)^3 = 1 + 21y + 210y^2 +...
A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
Rows begin with n=0:
  1;
  1, 14;
  1, 21,  210;
  1, 28,  378,  3220;
  1, 35,  595,  6475,  49910;
  1, 42,  861, 11396, 108402,  778596;
  1, 49, 1176, 18326, 207074, 1791930, 12198004;
  1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920;
  1, 63, 1953, 39585, 587727, 6783147, 62974371, 479497491, 3019005990; ...

Links

Crossrefs

Cf. A097179, A097182, A097183, A097184, A097185, A097186.

Programs

  • Mathematica
    Table[SeriesCoefficient[2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)), {x, 0,n}, {y,0,k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    {T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 2^n*(4^n -sum(j=0,n-1, T(n,j)/2^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}

Formula

G.f.: A(x, y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)).
G.f.: A(x, y) = A097183(x*y)/(1 - x*A097184(x*y)).

A097182 G.f. A(x) has the property that the first (n+1) terms of A(x)^(n+1) form the n-th row polynomial R_n(y) of triangle A097181 and satisfy R_n(1/2) = 8^n for all n>=0.

Original entry on oeis.org

1, 7, 21, 21, -63, -231, -15, 1521, 3073, -4319, -29631, -29631, 143361, 489345, -255, -3342591, -6684671, 9454081, 64553985, 64553985, -311689215, -1064175615, -4095, 7266627585, 14533263361, -20553129983, -140345589759, -140345589759, 677648531457, 2313636773889
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Examples

			A(x) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 - 15*x^6 +-...
For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the
n-th row polynomial R_n(y) of triangle A097181:
A^1 = {1, _7,  21,    21,    -63,    -231,      -15,     1521, ...}
A^2 = {1, 14, _91,   336,    609,    -462,    -5469,    -9516, ...}
A^3 = {1, 21, 210, _1288,   5103,   11655,     2160,   -85590, ...}
A^4 = {1, 28, 378,  3220, _18907,   77280,   199860,   153000, ...}
A^5 = {1, 35, 595,  6475,  49910, _283192,  1175190,  3282870, ...}
A^6 = {1, 42, 861, 11396, 108402,  778596, _4296034, 17959968, ...}
These row polynomials satisfy: R_n(1/2) = 8^n:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4.

Links

Crossrefs

Cf. A097181, A097183, A097184, A097185.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 16*x/(1-(1-2*x)^8) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series(16*x/(1-(1-2*x)^8), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[16*x/(1-(1-2*x)^8), {x,0,30}], x] (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    a(n)=polcoeff(16*x/(1-(1-2*x)^8)+x*O(x^n),n,x)
  • Sage
    def A097194_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(16*x/(1-(1-2*x)^8)).list()
A097194_list(30) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 16*x/(1-(1-2*x)^8).

A097183 Main diagonal of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.

Original entry on oeis.org

1, 14, 210, 3220, 49910, 778596, 12198004, 191682920, 3019005990, 47633205620, 752604648796, 11904837171864, 188493255221180, 2986893121197160, 47363590921840680, 751502309293205456, 11930099160029636614
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A097181, A097182, A097184.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-16*x)^(-7/8) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series((1-16*x)^(-7/8), x, n+1), x, n), n = 0 ..20); # G. C. Greubel, Sep 17 2019
  • Mathematica
    Table[FullSimplify[(n+1)*16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-16*x)^(-7/8), {x,0,20}], x] (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    a(n)=polcoeff(1/(1-16*x+x*O(x^n))^(7/8),n,x)
  • Sage
    def A097183_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-16*x)^(-7/8)).list()
A097183_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = 1/(1-16*x)^(7/8).
a(n) = (n+1)*A097184(n).
a(n) = (n+1) * 16^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014

A301271 Expansion of (1-16*x)^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440
Offset: 0

Views

Author

Seiichi Manyama, Jun 15 2018

Keywords

Links

Crossrefs

Cf. A003557, A007947.
(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

Programs

  • PARI
    N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))

Formula

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020

A138781 Triangle read by rows: coefficients of polynomials arising in the spontaneous magnetization of the anisotropic square lattice Ising model (see pp. 174-5 of the Guttmann reference).

Original entry on oeis.org

1, 2, 3, 2, 3, 16, 32, 16, 3, 4, 46, 200, 305, 200, 46, 4, 5, 100, 770, 2380, 3472, 2380, 770, 100, 5, 6, 185, 2230, 11600, 30240, 41244, 30240, 11600, 2230, 185, 6, 7, 308, 5362, 42140, 172795, 393008, 515332, 393008, 172795, 42140, 5362, 308, 7
Offset: 1

Views

Author

Emeric Deutsch, May 11 2008

Keywords

Comments

Row n has 2n-1 terms.
Sum of entries in row n is A097184(n-1).

Examples

			Triangle starts:
1;
2,3,2;
3,16,32,16,3;
4,46,200,305,200,46,4

References

  • A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.

Crossrefs

Cf. A097184.

Programs

  • Maple
    M:=(1-16*x*y/((1-x)^2*(1-y)^2))^(1/8): oneminusM:=simplify(series(1-M,x=0, 10)): for n to 7 do P[n]:=sort((1/2)*(y-1)^(2*n)*coeff(oneminusM,x,n)/y) end do: for n to 7 do seq(coeff(P[n],y,k),k=0..2*n-2) end do; # yields sequence in triangular form

Formula

The row generating polynomial P[n,y] of row n is defined by 1-M(x,y)=2*y*Sum(P[n,y]/(1-y)^(2n)*x^n, n=1..infinity), where M(x,y)=(1-16xy/[(1-x)^2*(1-y)^2])^(1/8).
Showing 1-5 of 5 results.

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