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A003114
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Number of partitions of n into parts 5k+1 or 5k+4.
(Formerly M0266)
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53
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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; internal format)
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OFFSET
| 0,5
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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 (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), 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 (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Apr 16 2006
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REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 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(e), 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. 107.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
A. V. Sills, Finite Rogers-Ramanujan type identities. Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
G. E. Andrews, Three aspects of partitions
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
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 (perry(AT)globalnet.co.uk), 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
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EXAMPLE
| a(9)=5 because we have [9], [6, 1, 1, 1], [4, 4, 1], [4, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].
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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 (deutsch(AT)duke.poly.edu), Feb 27 2006
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MATHEMATICA
| 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 *)
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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
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CROSSREFS
| Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
Cf. A188216 (least part k occurs at least k times).
Cf. A047209, A203776.
Sequence in context: A000607 A114372 A046676 * A185227 A026823 A025148
Adjacent sequences: A003111 A003112 A003113 * A003115 A003116 A003117
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Herman P. Robinson
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