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A070047 Number of partitions of n in which no part appears more than twice and no two parts differ by 1. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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

REFERENCES

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

LINKS

Reinhard Zumkeller and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms 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.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Wikipedia, Bailey pair

Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

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

EXAMPLE

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.

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-Fran├žois 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 *)

PROG

(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 */

(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

CROSSREFS

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

Sequence in context: A237832 A074500 A107237 * A101198 A034394 A058689

Adjacent sequences:  A070044 A070045 A070046 * A070048 A070049 A070050

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 09 2002

EXTENSIONS

Additional comments from Michael Somos, Dec 04 2002

STATUS

approved

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Last modified November 25 20:20 EST 2020. Contains 338627 sequences. (Running on oeis4.)