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 A003114 Number of partitions of n into parts 5k+1 or 5k+4. (Formerly M0266) 56

%I M0266

%S 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,

%T 79,91,102,117,131,149,167,189,211,239,266,299,333,374,415,465,515,

%U 575,637,709,783,871,961,1065,1174,1299,1429,1579,1735,1913,2100,2311,2533,2785

%N Number of partitions of n into parts 5k+1 or 5k+4.

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

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

%C 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

%C 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))).

%C Coefficients in expansion of permanent of infinite tridiagonal matrix:

%C 1 1 0 0 0 0 0 0 ...

%C x 1 1 0 0 0 0 0 ...

%C 0 x^2 1 1 0 0 0 ...

%C 0 0 x^3 1 1 0 0 ...

%C 0 0 0 x^4 1 1 0 ...

%C ................... - _Vladeta Jovovic_, Jul 17 2004

%C 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

%C 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

%C 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

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

%D G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.

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

%D G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.

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

%D R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

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

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

%H T. D. Noe, <a href="/A003114/b003114.txt">Table of n, a(n) for n=0..1000</a>

%H G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>

%H P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/hep-th/0505097">Parafermionic derivation of Andrews-type multiple-sums</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Rogers-Ramanujan Identities.</a>

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

%F 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

%F 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

%F Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - _Michael Somos_, Oct 15 2008

%e From _Joerg Arndt_, Dec 27 2012: (Start)

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

%e [ 1] [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

%e [ 2] [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]

%e [ 3] [ 4 4 1 1 1 1 1 1 1 1 ]

%e [ 4] [ 4 4 4 1 1 1 1 ]

%e [ 5] [ 4 4 4 4 ]

%e [ 6] [ 6 1 1 1 1 1 1 1 1 1 1 ]

%e [ 7] [ 6 4 1 1 1 1 1 1 ]

%e [ 8] [ 6 4 4 1 1 ]

%e [ 9] [ 6 6 1 1 1 1 ]

%e [10] [ 6 6 4 ]

%e [11] [ 9 1 1 1 1 1 1 1 ]

%e [12] [ 9 4 1 1 1 ]

%e [13] [ 9 6 1 ]

%e [14] [ 11 1 1 1 1 1 ]

%e [15] [ 11 4 1 ]

%e [16] [ 14 1 1 ]

%e [17] [ 16 ]

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

%e [ 1] [ 7 5 3 1 ]

%e [ 2] [ 8 5 3 ]

%e [ 3] [ 8 6 2 ]

%e [ 4] [ 9 5 2 ]

%e [ 5] [ 9 6 1 ]

%e [ 6] [ 9 7 ]

%e [ 7] [ 10 4 2 ]

%e [ 8] [ 10 5 1 ]

%e [ 9] [ 10 6 ]

%e [10] [ 11 4 1 ]

%e [11] [ 11 5 ]

%e [12] [ 12 3 1 ]

%e [13] [ 12 4 ]

%e [14] [ 13 3 ]

%e [15] [ 14 2 ]

%e [16] [ 15 1 ]

%e [17] [ 16 ]

%e (End)

%p 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

%t CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* From Jean-François Alcover, Apr 8 2011, after Emeric Deutsch *)

%o (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 */

%o (Haskell)

%o a003114 = p a047209_list where

%o p _ 0 = 1

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Jan 05 2011

%o (Haskell)

%o a003114 = p 1 where

%o p _ 0 = 1

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

%o -- _Reinhard Zumkeller_, Feb 19 2013

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

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

%Y Cf. A047209, A203776.

%K easy,nonn,nice,changed

%O 0,5

%A _N. J. A. Sloane_, Herman P. Robinson

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Last modified May 19 16:38 EDT 2013. Contains 225433 sequences.