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 1-4 of 4 results.

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.

Original entry on oeis.org

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Links

Crossrefs

Main diagonal of A296068.
Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

A106337 Number of ways of writing n as the sum of n triangular numbers.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 82, 253, 757, 2173, 6341, 18888, 56266, 167324, 499773, 1499059, 4503557, 13546893, 40824379, 123233868, 372472353, 1127080252, 3414310032, 10353722919, 31425764410, 95463814056, 290222666436, 882954212908, 2688037654049, 8188468874808
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

Number of compositions of n into n triangular numbers with 0's allowed. a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - Alois P. Heinz, Jul 31 2017
The radius of convergence is equal to A106335. - Vaclav Kotesovec, Nov 15 2017

Examples

			G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...

Links

Crossrefs

Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.
Main diagonal of A286180.

Programs

  • Maple
    b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017
  • Mathematica
    QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017 *)
  • PARI
    {a(n)=local(X); if(n<1,1,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n,n))}

Formula

Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).
a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - Alois P. Heinz, Aug 01 2017

Extensions

a(0) changed to 1 by Alois P. Heinz, Jul 31 2017

A296043 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.

Original entry on oeis.org

1, 1, -1, -5, -1, 31, 65, -90, -641, -644, 3329, 11386, -1471, -87021, -164634, 317935, 1881471, 1418719, -11370760, -33937951, 17468929, 294971868, 468897758, -1304743033, -6275603903, -2804572819, 42665919997, 109181454826, -106020803386, -1063546684834, -1362993953395
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Links

Crossrefs

Main diagonal of A296067.
Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A296044, A296045.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^n, {x, 0, n}], {n, 0, 30}]

A192540 G.f.: A(x) = Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (-x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 255, 960, 3707, 14597, 58382, 236522, 968597, 4003061, 16674858, 69936760, 295092057, 1251747436, 5334958079, 22834290248, 98108081192, 422986894605, 1829443421394, 7935301625600, 34510975557383, 150456011512671, 657415433062780
Offset: 1

Views

Author

Paul D. Hanna, Jul 03 2011

Keywords

Comments

Related q-series: Sum_{n>=0} (-q)^(n*(n+1)/2) = q^(-1/8)*eta(q)*eta(q^4)/eta(q^2) is a g.f. of A106459.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 + ...
The g.f. A = A(x) satisfies the following relations:
(1) A = x/(1 - A - A^3 + A^6 + A^10 - A^15 - A^21 + A^28 + A^36 + ...).
(2) A = x/((1-A)*(1+A^2)* (1-A^2)*(1+A^4)* (1-A^3)*(1+A^6)* (1-A^4)*(1+A^8)*...).
(3) A = x/((1-A)*(1-A^4)* (1-A^3)*(1-A^8)* (1-A^5)*(1-A^12)* (1-A^7)*(1-A^16)*...).
(4) A = x*(1+A)/(1-A^2)* (1+A^3)/(1-A^4)* (1+A^5)/(1-A^6) * (1+A^7)/(1-A^8)*...
(5) A = x*(1-A^2)/(1-A)* (1-A^6)/(1-A^2)* (1-A^10)/(1-A^3)* (1-A^14)/(1-A^4)*...
(6) A = x*exp(A/(1-A) - A^2/(2*(1+A^2)) + A^3/(3*(1-A^3)) - A^4/(4*(1+A^4)) + ...).
(7) A = x*exp(A + A^2/2 + 4*A^3/3 + 5*A^4/4 + 6*A^5/5 +...+ A113184(n)*A^n/n + ...).

Links

Crossrefs

Cf. A106459, A006950, A296045.

Programs

  • Maple
    nmax:=27: with(gfun): f := proc(x): x*add((-x)^(n*(n+1)/2),n=0..nmax) end: S:=series(f(x),x,nmax): g:= seriestoseries(S,'revogf'): seq(coeftayl (g,x=0,n),n=1..nmax); # Johannes W. Meijer, Jul 04 2011
  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x*EllipticTheta[2, 0, Sqrt[-x]] / (2*(-x)^(1/8)), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Aug 17 2015 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 8*(s/Sqrt[2*Pi*(77 - 8*(-s)^(7/8) *s*(Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-s]] / r))])} /. FindRoot[{2*r == -(-s)^(7/8)*EllipticTheta[2, 0, Sqrt[-s]], 2*(-s)^(11/8)*Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-s]] == 7*r}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
  • PARI
    {a(n)=polcoeff(serreverse(x*sum(m=0,sqrtint(2*n)+1,(-x)^(m*(m+1)/2)+x*O(x^n))),n)}
  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n,(1 - A^m)*(1 + A^(2*m))+x*O(x^n)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n\2,(1 - A^(2*m-1))*(1 - A^(4*m))+x*O(x^n)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n\2,(1 + A^(2*m-1))/(1 - A^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n,(1 - A^(4*m-2))/(1 - A^m+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=x*exp(sum(m=1, n, -(-A+x*O(x^n))^m/(1+(-A)^m)/m))); polcoeff(A, n)}
  • PARI
    {a(n)=if(n<1,0,(1/n)*polcoeff(x/prod(k=1,n,(1-x^k)*(1+x^(2*k)+x*O(x^n)))^n,n))}
  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(sum(m=1,n, A^m*sumdiv(m,d,(-1)^(m-d)*d)/m)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies:
(1) A(x) = x/[Sum_{n>=0} (-A(x))^(n*(n+1)/2)].
(2) A(x) = x/[Product_{n>=1} (1 - A(x)^n)*(1 + A(x)^(2*n))].
(3) A(x) = x/[Product_{n>=1} (1 - A(x)^(2*n-1))*(1 - A(x)^(4*n))].
(4) A(x) = x* Product_{n>=1} (1 + A(x)^(2*n-1))/(1 - A(x)^(2*n)).
(5) A(x) = x* Product_{n>=1} (1 - A(x)^(4*n-2))/(1 - A(x)^n).
(6) A(x) = x* exp( Sum_{n>=1} -(-A(x))^n/(n*(1 + (-A(x))^n)) ).
(7) A(x) = x* exp( Sum_{n>=1} A(x)^n*Sum_{d|n} (-1)^(n-d)*d/n ).
a(n) = [x^n] (1/n)*x/[Product_{k>=1} (1 - x^k)*(1 + x^(2*k))]^n for n >= 1.
a(n) ~ c * d^n / n^(3/2), where d = 4.6257905683677649210878404538251898489748116820946869227688637924996..., c = 0.1001072494040204029591345793571534412084516176488795... . - Vaclav Kotesovec, Aug 17 2015
Showing 1-4 of 4 results.

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