|
| |
|
|
A003106
|
|
Number of partitions of n into parts 5k+2 or 5k+3.
(Formerly M0261)
|
|
82
|
|
|
|
1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 38, 45, 50, 58, 64, 75, 82, 95, 105, 120, 133, 152, 167, 190, 210, 237, 261, 295, 324, 364, 401, 448, 493, 551, 604, 673, 739, 820, 899, 997, 1091, 1207, 1321, 1457, 1593, 1756, 1916, 2108, 2301
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
COMMENTS
|
Expansion of Rogers-Ramanujan function H(x) in powers of x.
Also number of partitions of n such that the number of parts is greater by one than the smallest part. - Vladeta Jovovic, Mar 04 2006
Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then there are exactly k-1 parts equal to k. Example: a(10)=4 because we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then k occurs at least k+1 times. Example: a(10)=4 because we have [2, 2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that the smallest part is larger than the number of parts. Example: a(10)=4 because we have [10], [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch, Apr 09 2006
Also number of partitions into distinct parts where parts differ by at least 2 and with minimal part >= 2, a(0)=1 because the condition is void for the empty list. - Joerg Arndt, Jan 04 2011
The g.f. is the special case D=2 of Sum_{n>=0} x^(D*n*(n+1)/2) / Product_{k=1..n} (1-x^k), the g.f. or partitions into distinct parts where the difference between successive parts is >= D and the minimal part >= D. - Joerg Arndt, Mar 31 2014
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[2](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109699 and A109698. - Vaclav Kotesovec, Jan 21 2017
|
|
|
REFERENCES
|
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities
|
|
|
FORMULA
|
The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. for the sequence.
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2 + 2*k) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos, Oct 15 2008
From Joerg Arndt, Oct 10 2012: (Start)
Bill Gosper gives (message to the math-fun mailing list, Oct 07 2012)
prod(k>=0, [0 , a; q^k, 1]) = [0, X(a,q); 0, Y(a,q)] where
X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and
Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ).
Set a=q to obtain prod(k>=0, [0 , a; q^k, 1]) = [0, q*H(q); 0, G(q)] where
H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114.
Bill Gosper and N. J. A. Sloane give (message to math-fun, Oct 10 2012)
prod(k>=0, [0 , a*q^k; 1, 1]) = [U(a,q), U(a,q); V(a,q), V(a,q)] where
U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^k) ) and
V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^k) ).
Set a=1 to obtain prod(k>=0, [0 , q^k; 1, 1]) = [H(q), H(q); G(q), G(q)].
(End)
Expansion of f(-x^5) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, May 06 2015
Expansion of f(-x, -x^4) / f(-x) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ sqrt((sqrt(5)-1)/5) * exp(2*Pi*sqrt(n/15)) / (2^(3/2) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(60*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 24 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017
|
|
|
EXAMPLE
|
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...
G.f. = q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 3*q^551 + ...
From Joerg Arndt, Dec 27 2012: (Start)
The a(18)=15: the partitions of 18 where all parts are 2 or 3 (mod 5) are
[ 1] [ 2 2 2 2 2 2 2 2 2 ]
[ 2] [ 3 3 2 2 2 2 2 2 ]
[ 3] [ 3 3 3 3 2 2 2 ]
[ 4] [ 3 3 3 3 3 3 ]
[ 5] [ 7 3 2 2 2 2 ]
[ 6] [ 7 3 3 3 2 ]
[ 7] [ 7 7 2 2 ]
[ 8] [ 8 2 2 2 2 2 ]
[ 9] [ 8 3 3 2 2 ]
[10] [ 8 7 3 ]
[11] [ 8 8 2 ]
[12] [ 12 2 2 2 ]
[13] [ 12 3 3 ]
[14] [ 13 3 2 ]
[15] [ 18 ]
(End)
From Wolfdieter Lang, Oct 29 2016: (Start)
The a(18)=15 partitions of 18 without part 1 and parts differing by at least 2 are:
[18]; [16,2], [15,3], [14,4], [13,5], [12,6], [11,7], [10,8]; [12,4,2], [11,5,2], [10,6,2], [9,7,2],[10,5,3], [9,6,3], [8,6,4]. The semicolon separates different number of parts. The maximal number of parts is A259361(18) = 3. (End)
|
|
|
MAPLE
|
g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)), j=1..15): gser:=series(g, x=0, 66): seq(coeff(gser, x, n), n=0..63); # Emeric Deutsch, Apr 09 2006
|
|
|
MATHEMATICA
|
max = 63; g[x_] := 1/Product[(1-x^(5j-2))*(1-x^(5j-3)), {j, 1, Floor[max/4]}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, -1, -1, 0, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
|
|
|
PROG
|
(PARI) {a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1), n))}; /* Michael Somos, Oct 15 2008 */
(Haskell)
a003106 = p a047221_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 30 2012
|
|
|
CROSSREFS
|
Cf. A003114.
Cf. A047221, A219607.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Sequence in context: A319169 A050365 A029026 * A185228 A026824 A025149
Adjacent sequences: A003103 A003104 A003105 * A003107 A003108 A003109
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
N. J. A. Sloane, Herman P. Robinson
|
|
|
STATUS
|
approved
|
| |
|
|
