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A003114 Number of partitions of n into parts 5k+1 or 5k+4.
(Formerly M0266)
65
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Expansion of Rogers-Ramanujan function G(x) in powers of x.

Same as number of partitions into distinct parts where the difference between successive parts is >= 2.

As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).

Coefficients in expansion of permanent of infinite tridiagonal matrix:

1 1 0 0 0 0 0 0 ...

x 1 1 0 0 0 0 0 ...

0 x^2 1 1 0 0 0 ...

0 0 x^3 1 1 0 0 ...

0 0 0 x^4 1 1 0 ...

................... - Vladeta Jovovic, Jul 17 2004

Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004

Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006

Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006

a(n) = number of NW partitions of n, for n >= 1; see A237981.

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.

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, Three aspects of partitions

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

A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities.

FORMULA

G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).

The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014

G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004

G.f.: (Product_{k>0} 1+x^(2k))*(Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^(4i))). - Michael Somos, Oct 19 2006

Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008

EXAMPLE

From Joerg Arndt, Dec 27 2012: (Start)

The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are

[ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

[ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]

[ 3]  [ 4 4 1 1 1 1 1 1 1 1 ]

[ 4]  [ 4 4 4 1 1 1 1 ]

[ 5]  [ 4 4 4 4 ]

[ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ]

[ 7]  [ 6 4 1 1 1 1 1 1 ]

[ 8]  [ 6 4 4 1 1 ]

[ 9]  [ 6 6 1 1 1 1 ]

[10]  [ 6 6 4 ]

[11]  [ 9 1 1 1 1 1 1 1 ]

[12]  [ 9 4 1 1 1 ]

[13]  [ 9 6 1 ]

[14]  [ 11 1 1 1 1 1 ]

[15]  [ 11 4 1 ]

[16]  [ 14 1 1 ]

[17]  [ 16 ]

The a(16)=17 partitions of 16 where successive parts differ by at least 2 are

[ 1]  [ 7 5 3 1 ]

[ 2]  [ 8 5 3 ]

[ 3]  [ 8 6 2 ]

[ 4]  [ 9 5 2 ]

[ 5]  [ 9 6 1 ]

[ 6]  [ 9 7 ]

[ 7]  [ 10 4 2 ]

[ 8]  [ 10 5 1 ]

[ 9]  [ 10 6 ]

[10]  [ 11 4 1 ]

[11]  [ 11 5 ]

[12]  [ 12 3 1 ]

[13]  [ 12 4 ]

[14]  [ 13 3 ]

[15]  [ 14 2 ]

[16]  [ 15 1 ]

[17]  [ 16 ]

(End)

MAPLE

g:=sum(x^(k^2)/product(1-x^j, j=1..k), k=0..10): gser:=series(g, x=0, 65): seq(coeff(gser, x, n), n=0..60); - Emeric Deutsch, Feb 27 2006

MATHEMATICA

CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-Fran├žois Alcover, Apr 08 2011, after Emeric Deutsch *)

Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 24}] (* _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(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))} /* Michael Somos, Oct 15 2008 */

(Haskell)

a003114 = p a047209_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, Jan 05 2011

(Haskell)

a003114 = p 1 where

   p _ 0 = 1

   p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m

-- Reinhard Zumkeller, Feb 19 2013

CROSSREFS

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.

Cf. A188216 (least part k occurs at least k times).

Cf. A047209, A203776, A237981.

Sequence in context: A000607 A114372 A046676 * A185227 A217569 A026823

Adjacent sequences:  A003111 A003112 A003113 * A003115 A003116 A003117

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane, Herman P. Robinson

STATUS

approved

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Last modified September 2 10:05 EDT 2014. Contains 246350 sequences.