A122129 - OEIS

A122129

Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).

%I #59 Dec 22 2024 16:29:38

%S 1,1,1,2,3,4,5,7,9,12,15,19,24,30,37,46,57,69,84,102,123,148,177,211,

%T 252,299,353,417,491,576,675,789,920,1071,1244,1442,1670,1929,2224,

%U 2562,2946,3381,3876,4437,5072,5791,6602,7517,8551,9714,11021,12493,14145

%N Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).

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

%C a(n) = number of SE partitions of n, for n >= 1; see A237981. - _Clark Kimberling_, Mar 19 2014

%C 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

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

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

%C From _Gus Wiseman_, Feb 19 2022: (Start)

%C This appears to be the number of integer partitions of n with every other pair of adjacent parts strictly decreasing, as in the pattern a > b >= c > d >= e for a partition (a, b, c, d, e). For example, the a(1) = 1 through a(9) = 12 partitions are:

%C (1) (2) (3) (4) (5) (6) (7) (8) (9)

%C (21) (31) (32) (42) (43) (53) (54)

%C (211) (41) (51) (52) (62) (63)

%C (311) (321) (61) (71) (72)

%C (411) (322) (422) (81)

%C (421) (431) (432)

%C (511) (521) (522)

%C (611) (531)

%C (3221) (621)

%C (711)

%C (4221)

%C (32211)

%C The even-length case is A351008. The odd-length case appears to be A122130. Swapping strictly and weakly decreasing relations appears to give A122135. The alternately unequal and equal case is A351006, strict A035457, opposite A351005, even-length A351007. (End)

%C Wiseman's first conjecture above was proved by Gordon, Theorem 7. For two other combinatorial interpretations of this sequence see Connor, Proposition 1. - _Peter Bala_, Dec 22 2024

%D 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)

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

%H Alois P. Heinz, <a href="/A122129/b122129.txt">Table of n, a(n) for n = 0..1000</a>

%H Willard G. Connor, <a href="https://doi.org/10.2307/1997097">Partition Theorems Related to Some Identities of Rogers and Watson</a>, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.

%H Basil Gordon, <a href="https://doi.org/10.1215/S0012-7094-65-03278-3">Continued fractions of the Rogers-Ramanujan type</a>, Duke Math. J. 32 (1965), 741-748. MR 32 # 1477.

%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/0097-3165(79)90005-0">Some partition theorems of the Rogers-Ramanujan type</a>, 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]

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H George N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.274">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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, ...].

%F 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.

%F 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

%F 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

%F 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

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

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

%F 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

%F a(n) is the number of partitions of n into parts that are either odd or == +-4 (mod 20). - _Michael Somos_, Jun 28 2015

%F a(n) ~ (3+sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015

%e 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

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

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

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

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

%p series(%,q,200); seriestolist(%); # _N. J. A. Sloane_, Mar 19 2012.

%p # second Maple program:

%p with(numtheory):

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

%p 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1]

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

%p end:

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jul 12 2013

%t 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-François Alcover_, Jan 10 2014, after _Alois P. Heinz_ *)

%t 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 *)

%t 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 *)

%o (PARI)

%o {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))};

%Y Cf. A000009, A000041, A000070, A096441, A122134, A351004.

%K nonn,easy

%O 0,4

%A _Michael Somos_, Aug 21 2006