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 1-3 of 3 results.

A106507 G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, -1, 1, -2, 3, -4, 5, -7, 10, -13, 16, -21, 28, -35, 43, -55, 70, -86, 105, -130, 161, -196, 236, -287, 350, -420, 501, -602, 722, -858, 1016, -1206, 1431, -1687, 1981, -2331, 2741, -3206, 3740, -4368, 5096, -5922, 6868, -7967, 9233, -10670, 12306, -14193
Offset: 0

Views

Author

Michael Somos, May 04 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present entry gives 1/psi(q).
For various G.f. versions see the reciprocals of the ones given in A010054. - Wolfdieter Lang, Jul 05 2016

Examples

			G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 + ...
G.f. of B(q) =  A(q^8) / q = 1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.

Links

Crossrefs

Cf. A006950, A010054, A106336, A233758, A233759.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, May 28 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)] , {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
(QPochhammer[x, x^2, 1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))};

Formula

Expansion of 1 / psi(x) in powers of x where psi() is a Ramanujan theta function, which is Jacobi's theta_2(0, sqrt(x))/(2*x^(1/8)) function. See, e.g., the Eric Weisstein link.
Expansion of q^(1/8) * eta(q) / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -1, 1, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4 * (w^4 + 4*v^4) - v^6*w^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2*u6^3 + u2^2*u3^3 - u3^3*u6^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^3*u6^2 + 3*u1^3*u2^2 - u2^3*u3*u6.
G.f.: Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k in Z} x^((4k + 1)^2)).
G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan] - Michael Somos, Jul 21 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A015128. - Michael Somos, Nov 01 2008
a(n) = (-1)^n * A006950(n). Convolution inverse of A010054.
Series reversion of A106336. - Michael Somos, May 10 2012
a(2*n) = A233758(n). a(2*n + 1) = - A233759(n). - Michael Somos, Nov 05 2015
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(2*k)). - Michael Somos, Nov 08 2015
G.f.: (x; x^2){1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017

Extensions

Definition changed by N. J. A. Sloane, Aug 14 2007

A233758 Bisection of A006950 (the even part).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236, 350, 501, 722, 1016, 1431, 1981, 2741, 3740, 5096, 6868, 9233, 12306, 16357, 21581, 28394, 37128, 48406, 62777, 81182, 104494, 134131, 171467, 218607, 277691, 351841, 444314, 559727, 703002, 880896, 1100775
Offset: 1

Views

Author

Omar E. Pol, Jan 11 2014

Keywords

Comments

See Zaletel-Mong paper, page 14, FIG. 11: C2a is this sequence, C2b is A233759, C2c is A015128.

Links

Crossrefs

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233759.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 2, 2 n - 2];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A233969 Partial sums of A006950.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 34, 47, 63, 84, 112, 147, 190, 245, 315, 401, 506, 636, 797, 993, 1229, 1516, 1866, 2286, 2787, 3389, 4111, 4969, 5985, 7191, 8622, 10309, 12290, 14621, 17362, 20568, 24308, 28676, 33772, 39694, 46562, 54529, 63762, 74432, 86738
Offset: 0

Views

Author

Omar E. Pol, Jan 12 2014

Keywords

Comments

The first three columns of A211970 are A211971, A000041, A006950, so for k = 0..2, the partial sums of column k of A211970 give: A015128, A000070, this sequence.

Links

Crossrefs

Cf. A000041, A000070, A015128, A195825, A195826, A195836, A210843, A211970, A211971, A233758, A233759.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 12 2014
  • Mathematica
    Accumulate[CoefficientList[Series[x*QPochhammer[-1/x, x^2]/((1 + x) * QPochhammer[x^2]), {x, 0, 50}], x]] (* Vaclav Kotesovec, Oct 27 2016 *)

Formula

a(n) ~ exp(Pi*sqrt(n/2))/(2*Pi*sqrt(n)). - Vaclav Kotesovec, Oct 27 2016
Showing 1-3 of 3 results.

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