A070047 Number of partitions of n in which no part appears more than twice and no two parts differ by 1.
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, 156, 196, 210, 262, 283, 348, 376, 459, 497, 600, 651, 781, 849, 1009, 1097, 1298, 1413, 1660, 1807, 2113, 2302, 2676, 2916, 3377, 3681, 4242, 4623, 5309, 5787, 6619
Offset: 0
Keywords
Examples
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 + ... 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 + ... a(6)=5 because we have 6,51,42,411,33.
References
- D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..120 from Reinhard Zumkeller)
- G. E. Andrews and R. P. Lewis, An algebraic identity of F. H. Jackson and its implications for partitions, Discrete Math., 232 (2001), 77-83. see equations (3.1) and (3.2)
- Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, Electronic Journal of Combinatorial Number Theory, Volume 15 #A23.
- Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- Wikipedia, Bailey pair
- Mingjia Yang and Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.
Programs
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Haskell
a070047 n = p 1 n where p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+1) m q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m -- Reinhard Zumkeller, Nov 12 2011
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1) +add(b(n-i*j, i-2), j=1..min(n/i, 2)))) end: a:= n-> b(n, n): seq(a(n), n=0..60); # Alois P. Heinz, Jan 18 2013 -
Mathematica
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 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *) 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 *) 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 *) -
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 */
Formula
Expansion of phi(-x^3) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Jun 02 2011
Expansion of q^(1/24) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Dec 04 2002
Euler transform of period 6 sequence [ 1, 1, -1, 1, 1, 0, ...]. - Michael Somos, Dec 04 2002
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.
G.f.: Prod_{k>0} (1 - x^(6*k - 3))^2 * (1 - x^(6*k)) / (1 - x^k).
G.f.: Prod_{n>0}[(1-q^(6n-3))/[(1-q^(3n-2))(1-q^(3n-1))]]. - Emeric Deutsch, May 19 2008
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
G.f.: Product_{n >= 1} (1 + q^n + q^(2*n))/(1 + q^(3*n)). - Peter Bala, Jan 04 2025
Extensions
Additional comments from Michael Somos, Dec 04 2002
Comments