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Search: id:A001979
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| A001979 |
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Number of partitions of floor(7n/2) into n nonnegative integers each no more than 7.
(Formerly M3389 N1369)
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+0
3
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| 1, 1, 4, 10, 24, 49, 94, 169, 289, 468, 734, 1117, 1656, 2385, 3370, 4672, 6375, 8550, 11322, 14800, 19138, 24460, 30982, 38882, 48417, 59779, 73316, 89291, 108108, 130053, 155646, 185258, 219489, 258735, 303748, 355034, 413442, 479500, 554256
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Also, the dimension of the vector space of homogeneous covariants of degree n for the binary form of degree 7. To calculate the dimension one uses the Sylvester-Cayley formula. - Leonid Bedratyuk (bedratyuk(AT)ief.tup.km.ua), Dec 06 2006
In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2) involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.
Springer, T.A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, (1977).
Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).
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LINKS
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A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.
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FORMULA
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Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)(1-x^7z)), where w=floor(7n/2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008
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EXAMPLE
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a(14)=2385, a(15)=3370, a(16)=4672, a(17)=6375.
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MAPLE
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a(n+1) = subs({x=1}, convert(series((product('1-x^i', 'i'=8..7+n)/product('1-x^k', 'k'=2..n)), x, trunc(7*n/2)+1), polynom)); - Leonid Bedratyuk (bedratyuk(AT)ief.tup.km.ua), Dec 06 2006
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PROGRAM
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(PARI) f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)*(1-x^7*z)); n=450; p=subst(subst(f, x, x+x*O(x^n)), z, z+z*O(z^n)); for(d=0, 60, w=floor(7*d/2); print1(polcoeff(polcoeff(p, w), d)", ")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008
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CROSSREFS
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Sequence in context: A083168 A143696 A058514 this_sequence A128516 A022569 A093831
Adjacent sequences: A001976 A001977 A001978 this_sequence A001980 A001981 A001982
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008
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