%I #56 Oct 27 2023 19:57:55

%S 1,1,2,1,3,3,5,5,8,8,12,12,19,19,27,28,39,41,55,58,77,82,106,113,145,

%T 156,196,210,262,283,348,376,459,497,600,651,781,849,1009,1097,1298,

%U 1413,1660,1807,2113,2302,2676,2916,3377,3681,4242,4623,5309,5787,6619

%N Number of partitions of n in which no part appears more than twice and no two parts differ by 1.

%C Coefficients in expansion of permanent of infinite tridiagonal matrix: matrix([[1, x, 0, 0, 0, ...], [1+x, 1, x^2, 0, 0, ...], [0, 1+x^2, 1, x^3, 0, ...], [0, 0, 1+x^3, 1, x^4, ...], ...]). - _Vladeta Jovovic_, Jul 18 2004

%C Number of partitions of n into non-multiples of 3 in which no two parts differ by 1 (see the Andrews-Lewis reference). Example: a(6)=5 because we have 51,42,411,222,111111. - _Emeric Deutsch_, May 19 2008

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

%D D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

%H Alois P. Heinz, <a href="/A070047/b070047.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..120 from Reinhard Zumkeller)

%H G. E. Andrews and R. P. Lewis, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00295-8">An algebraic identity of F. H. Jackson and its implications for partitions</a>, Discrete Math., 232 (2001), 77-83. see equations (3.1) and (3.2)

%H Bin Lan and James A. Sellers, <a href="http://www.emis.de/journals/INTEGERS/papers/p23/p23.pdf">Properties of a Restricted Binary Partition Function a la Andrews and Lewis</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A23.

%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey_pair">Bailey pair</a>

%H Mingjia Yang, Doron Zeilberger, <a href="https://arxiv.org/abs/1910.08989">Systematic Counting of Restricted Partitions</a>, arXiv:1910.08989 [math.CO], 2019.

%F Expansion of phi(-x^3) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Jun 02 2011

%F Expansion of q^(1/24) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q. - _Michael Somos_, Dec 04 2002

%F Euler transform of period 6 sequence [ 1, 1, -1, 1, 1, 0, ...]. - _Michael Somos_, Dec 04 2002

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g is the g.f. of A233006.

%F G.f.: Prod_{k>0} (1 - x^(6*k - 3))^2 * (1 - x^(6*k)) / (1 - x^k).

%F G.f.: Prod_{n>0}[(1-q^(6n-3))/[(1-q^(3n-2))(1-q^(3n-1))]]. - _Emeric Deutsch_, May 19 2008

%F a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)/2)) / sqrt(3*(24*n-1)) ~ exp(Pi*sqrt(n/3)) / (2*3^(3/4)*n^(3/4)) * (1 - (3*sqrt(3)/(8*Pi) + Pi/(48*sqrt(3)))/sqrt(n) + (Pi^2/13824 - 45/(128*Pi^2) + 5/128)/n). - _Vaclav Kotesovec_, Sep 02 2015, extended Jan 11 2017

%e G.f. = 1 + x + 2*x^2 + x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + 12*x^10 + ...

%e G.f. = 1/q + q^23 + 2*q^47 + q^71 + 3*q^95 + 3*q^119 + 5*q^143 + 5*q^167 + 8*q^191 + ...

%e a(6)=5 because we have 6,51,42,411,33.

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1) +add(b(n-i*j, i-2), j=1..min(n/i, 2))))

%p end:

%p a:= n-> b(n, n):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 18 2013

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Product[ (1 - x^(6 k - 3))^2 (1 - x^(6 k)), {k, Ceiling[ n/6]}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]]; (* _Michael Somos_, Jun 02 2011 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Dec 03 2013 *)

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-2], {j, 1, Min[n/i, 2]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Mar 04 2015, after _Alois P. Heinz_ *)

%t nmax = 100; CoefficientList[Series[Product[1 / ( (1-x^(3*k-2)) * (1-x^(3*k-1)) * (1 + x^(3*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))}; /* _Michael Somos_, Jun 02 2011 */

%o (Haskell)

%o a070047 n = p 1 n where

%o p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+1) m

%o q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m

%o -- _Reinhard Zumkeller_, Nov 12 2011

%Y Cf. A233006, A000700, A108961, A108962, A271661, A280937, A280938.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, May 09 2002

%E Additional comments from _Michael Somos_, Dec 04 2002