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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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2
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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
| 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
| 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
| a(14)=2385, a(15)=3370, a(16)=4672, a(17)=6375.
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MAPLE
| 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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PROG
| (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
| Sequence in context: A083168 A143696 A058514 * A128516 A022569 A093831
Adjacent sequences: A001976 A001977 A001978 * A001980 A001981 A001982
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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
| Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008
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