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.

A097184 G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.

Original entry on oeis.org

1, 7, 70, 805, 9982, 129766, 1742572, 23960365, 335445110, 4763320562, 68418604436, 992069764322, 14499481170860, 213349508656940, 3157572728122712, 46968894330825341, 701770538825272742, 10526558082379091130, 158452400608443161220
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A097181, A097182, A097183.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (1-(1-16*x)^(1/8))/(2*x) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series((1-(1-16*x)^(1/8))/(2*x), x, n+2), x, n), n = 0..20); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[(1-(1-16*x)^(1/8))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
Table[FullSimplify[16^n*Gamma[n+7/8]/(Gamma[7/8]*Gamma[n+2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)
  • PARI
    a(n)=polcoeff((1-(1-16*x+x^2*O(x^n))^(1/8))/(2*x),n,x)
  • Sage
    def A097184_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1-(1-16*x)^(1/8))/(2*x)).list()
A097184_list(20) # G. C. Greubel, Sep 17 2019

Formula

G.f.: A(x) = (1-(1-16*x)^(1/8))/(2*x).
G.f.: A(x) = (1/x)*(series reversion of x/A097182(x)).
a(n) = A097183(n)/(n+1).
D-finite with recurrence: (n+1)*a(n) +2*(-8*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 16^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
a(n) ~ 16^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Feb 09 2014

Extensions

More terms from Vincenzo Librandi, Feb 10 2014

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

A097185 Row sums 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, 15, 232, 3627, 57016, 899298, 14216560, 225110307, 3568890328, 56635884470, 899474459280, 14294357356110, 227286593929136, 3615608476770340, 57538659207907552, 915981394162628387, 14586262906867731096
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2004

Keywords

Links

Crossrefs

Cf. A097181, A097182.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/((1-16*x) + (1-16*x)^(7/8)) )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series(2/((1-16*x) + (1-16*x)^(7/8)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[2/((1-16*x) +(1-16*x)^(7/8)), {x,0,30}], x] (* G. C. Greubel, Sep 17 2019 *)
  • PARI
    a(n)=polcoeff(2/((1-16*x)+(1-16*x+x*O(x^n))^(7/8)),n,x)
  • Sage
    def A097185_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(2/((1-16*x) + (1-16*x)^(7/8))).list()
A097185_list(30) # G. C. Greubel, Sep 17 2019

Formula

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

A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

Original entry on oeis.org

1, 1, 6, 1, 9, 42, 1, 12, 74, 308, 1, 15, 115, 595, 2310, 1, 18, 165, 1020, 4746, 17556, 1, 21, 224, 1610, 8722, 37730, 134596, 1, 24, 292, 2392, 14778, 73080, 299508, 1038312, 1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2004

Keywords

Comments

Row sums form A097180. Diagonal is A004982. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A048779, where the g.f.s satisfy: A077860(x*A048779(x)) = A048779(x).

Examples

			Row polynomials evaluated at y=1/2 equals powers of 4:
4^1 = 1 + 6/2;
4^2 = 1 + 9/2 + 42/2^2;
4^3 = 1 + 12/2 + 74/2^2 + 308/2^3;
4^4 = 1 + 15/2 + 115/2^2 + 595/2^3 + 2310/2^4;
where A077860(y)^(n+1) has the same initial terms as the n-th row:
A077860(y) = 1 +3*y +5*y^2 +5*y^3 +1*y^4 -7*y^5 -15*y^6 -15*y^7 +...
A077860(y)^2 = 1 + 6*y +...
A077860(y)^3 = 1 + 9*y + 42*y^2 +...
A077860(y)^4 = 1 + 12*y + 74*y^2 + 308*y^3 +...
A077860(y)^5 = 1 + 15*y + 115*y^2 + 595*y^3 + 2310*y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  42;
  1, 12,  74,  308;
  1, 15, 115,  595,  2310;
  1, 18, 165, 1020,  4746, 17556;
  1, 21, 224, 1610,  8722, 37730,  134596;
  1, 24, 292, 2392, 14778, 73080,  299508, 1038312;
  1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918; ...

Links

Crossrefs

Cf. A004982, A048779, A077860, A097180, A097181.

Programs

  • Mathematica
    Table[SeriesCoefficient[2*y/((1-8*x*y) +(2*y-1)*(1-8*x*y)^(3/4)), {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/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)).
G.f.: A(x, y) = A004982(x*y)/(1 - x*A048779(x*y)).
Showing 1-5 of 5 results.

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