login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function. 15
1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 7, 10, 12, 13, 14, 16, 21, 27, 32, 35, 38, 44, 54, 67, 78, 86, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 26 2012

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 07 2012

Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - Michael Somos, Jun 07 2012

Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014

Convolution inverse of A089802. - Michael Somos, Oct 18 2014

a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - Vaclav Kotesovec, Nov 08 2015

a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

EXAMPLE

G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...

G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...

MAPLE

A001082 := proc(n)

        if type(n, 'even') then

                n*(3*n-4)/4 ;

        else

                (n-1)*(3*n+1)/4 ;

        end if;

end proc:

A195838 := proc(n, k)

        option remember;

        local ks, a, j ;

        if A001082(k+1) > n then

                0 ;

        elif n <= 5 then

                return 1;

        elif k = 1 then

                a := 0 ;

                for j from 1 do

                        if A001082(j+1) <= n-1 then

                                a := a+procname(n-1, j) ;

                        else

                                break;

                        end if;

                end do;

                return a;

        else

                ks := A001082(k+1) ;

                (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;

        end if;

end proc:

A195848 := proc(n)

        A195838(n+1, 1) ;

end proc:

seq(A195848(n), n=0..60) ; # R. J. Mathar, Oct 07 2011

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* Michael Somos, Oct 18 2014 *)

a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Oct 18 2014 *)

nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 07 2012 */

From Omar E. Pol, Jun 10 2012: (Start)

(GWbasic)' A program with two A-numbers:

10 Dim A001082(100), A057077(100), a(100): a(0)=1

20 For n = 1 to 58: For j = 1 to n

30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))

40 Next j: Print a(n-1); : Next n (End)

CROSSREFS

Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.

Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.

Cf. A089802.

Sequence in context: A199332 A029085 A087875 * A099777 A221917 A131798

Adjacent sequences:  A195845 A195846 A195847 * A195849 A195850 A195851

KEYWORD

nonn

AUTHOR

Omar E. Pol, Sep 24 2011

EXTENSIONS

New sequence name from Michael Somos, Oct 18 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 22 01:06 EDT 2019. Contains 321406 sequences. (Running on oeis4.)