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 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 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

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