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A003106
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Number of partitions of n into parts 5k+2 or 5k+3.
(Formerly M0261)
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40
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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
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OFFSET
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0,7
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COMMENTS
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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
From Joerg Arndt, Oct 10 2012: (Start)
R. W. 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.
R. W. 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)
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities
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FORMULA
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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^(2k))*(Sum_{k>=0} x^(k^2+2k)/(Product_{i=1..k} 1-x^(4i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos, Oct 15 2008
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EXAMPLE
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1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^8 + 4*x^9 + 4*x^10 +...
q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + ...
From Joerg Arndt, Dec 27 2012: (Start)
The a(18)=15 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)
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MAPLE
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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
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MATHEMATICA
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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] (* From Jean-François Alcover, Nov 17 2011, after Emeric Deutsch *)
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PROG
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(PARI) {a(n) = local(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
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CROSSREFS
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Cf. A003114.
Cf. A047221, A219607.
Sequence in context: A058747 A050365 A029026 * A185228 A026824 A025149
Adjacent sequences: A003103 A003104 A003105 * A003107 A003108 A003109
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Herman P. Robinson
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STATUS
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approved
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