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A003106 Number of partitions of n into parts 5k+2 or 5k+3.
(Formerly M0261)
45
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) / prod(k=1..n,1-x^k) ), the g.f. or partitions into distinct part where the difference between successive parts is >= D and the minimal part >= D. - Joerg Arndt, Mar 31 2014

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

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

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^(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

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)

EXAMPLE

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)

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-Fran├žois Alcover, Nov 17 2011, after Emeric Deutsch *)

Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)

PROG

(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

CROSSREFS

Cf. A003114.

Cf. A047221, A219607.

Sequence in context: A034322 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

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.