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A006141
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Number of balanced partitions of n: last element is equal to number of elements.
(Formerly M0260)
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3
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1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,9
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COMMENTS
| Or, number of partitions of n in which number of largest parts is equal to largest part.
Related to Rogers-Ramanujan identities.
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REFERENCES
| Andrews, George E.; Baxter, R. J.; A motivated proof of the Rogers-Ramanujan identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| G.f.: 1 + sum(i=1, oo, x^i(i+2)/product(j=1, i, 1-x^j)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 04 2004
G.f.: Sum((x^(m^2)-x^(m*(m+1)))/Product(1-x^i, i=1..m), m=1..infinity).
G.f.: sum(n>=0, x^(n^2)/prod(k=1,n-1,1-x^k)) - Joerg Arndt, Jan 29 2011
a(n) = A003114(n)-A003106(n) = A039900(n)-A039889(n), (offset 1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 17 2004
Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z**4+2*z**5-z**3-z**8+3*z**10-z**7+z**9)/(1+z-z**4-2*z**3-z**8+z**10), but Michael Somos pointed out on Jan 22 2008 that this is false.
The number of partitions of n+1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f() is Ramanujan's theta function. - Michael Somos 22 Jan 2007
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EXAMPLE
| a(10) = 2 since 9 = 6 + 3. a(12) = 3 since 11 = 8 + 3 = 7 + 4.
{7, 2} is a balanced partition of 9 because the last element is 2 and the length is 2. The other is {3, 3, 3} so a(9) = 2.
q + q^4 + q^5 + q^6 + q^7 + q^8 + 2*q^9 + 2*q^10 + 3*q^11 + 3*q^12 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos Jan 22 2008 */
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CROSSREFS
| Cf. A047993.
Sequence in context: A025765 A029029 A025157 * A185229 A026825 A025150
Adjacent sequences: A006138 A006139 A006140 * A006142 A006143 A006144
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
Better description from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 06 2002
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