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A122129 Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))). 9
1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 30, 37, 46, 57, 69, 84, 102, 123, 148, 177, 211, 252, 299, 353, 417, 491, 576, 675, 789, 920, 1071, 1244, 1442, 1670, 1929, 2224, 2562, 2946, 3381, 3876, 4437, 5072, 5791, 6602, 7517, 8551, 9714, 11021, 12493, 14145 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.

a(n) = number of SE partitions of n, for n >= 1; see A237981.  Clark Kimberling, Mar 19 2014

In Watson 1937 page 275 he writes "Psi_0(1,q) = prod_1^oo (1+q^{2n}) G(q^8)" so this is the expansion in powers of q^2. - Michael Somos, Jun 28 2015

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

Rogers-Ramanujan functions: G(x) (see A003114), H(x) (A003106).

REFERENCES

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.7). MR0858826 (88b:11063)

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(a), p. 591.

Watson, G. N. (1937), "The Mock Theta Functions (2)", Proceedings of the London Mathematical Society, s2-42: 274-304, doi:10.1112/plms/s2-42.1.274

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33--37. MR0541341 (80j:05010). See Theorem 1. [From N. J. A. Sloane, Mar 19 2012]

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 20 sequence [ 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...].

Expansion of f(-x^2) * f(-x^20) / (f(-x) * f(-x^4,-x^16)) in powers of x where f(,) is the Ramanujan general theta function.

Expansion of f(x^3, x^7) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 28 2015

Expansion of f(-x^8, -x^12) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jun 28 2015

Expansion of G(x^4) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 28 2015

G.f.: Sum_{k>=0} x^k^2 / ((1 - x) * (1 - x^2) ... (1 - x^(2*k))).

G.f.: 1 / (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(20*k-4)) * (1 - x^(20*k-16))).

Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19). - N. J. A. Sloane, Mar 19 2012.

a(n) = number of partitions of n into parts that are either odd or == +/-4 (mod 20). - Michael Somos, Jun 28 2015

a(n) ~ (3+sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

EXAMPLE

Clark Kimberling's SE partition comment, n=6: the 5 SE partitions are [1,1,1,1,1,1] from the partitions 6 and 1^6; [1,1,1,2,1] from 5,1 and 2,1^4; [1,1,3,1] from 4,2 and 2^2,1^2; [2,3,1] from 3,2,1 and 3^2 and 2^3; and [1,2,2,1] from 4,1^2 and 3,1^3. - Wolfdieter Lang, Mar 20 2014

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

G.f. = 1/q + q^39 + q^79 + 2*q^119 + 3*q^159 + 4*q^199 + 5*q^239 + ...

MAPLE

f:=n->1/mul(1-q^(20*k+n), k=0..20);

f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19);

series(%, q, 200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012.

# second Maple program:

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, 1, 0,

       1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1]

      [1+irem(d, 20)], d=divisors(j)) *a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Jul 12 2013

MATHEMATICA

a[0] = 1; a[n_] := a[n] = Sum[Sum[d*{0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1}[[1+Mod[d, 20]]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Jan 10 2014, after Alois P. Heinz *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ x, x, 2 k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 28 2015 *)

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^4, x^20] QPochhammer[ x^16, x^20]), {x, 0, n}]; (* Michael Somos, Jun 28 2015 *)

PROG

a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n-k^2)))), n))};

CROSSREFS

Cf. A122135.

Sequence in context: A306385 A039853 A062188 * A280909 A003413 A100853

Adjacent sequences:  A122126 A122127 A122128 * A122130 A122131 A122132

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 21 2006

STATUS

approved

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Last modified March 31 02:36 EDT 2020. Contains 333135 sequences. (Running on oeis4.)