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A006463
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Convolve natural numbers with characteristic function of triangular numbers.
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3
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0, 0, 1, 2, 4, 6, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 45, 50, 55, 60, 65, 70, 76, 82, 88, 94, 100, 106, 112, 119, 126, 133, 140, 147, 154, 161, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 340, 350, 360
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
a(n) = length (i.e. number of elements minus 1) of longest chain in partition lattice Par(n). Par(n) is the set of partitions of n under "dominance order": partition P is <= partition Q iff the sum of the largest k parts of P is <= the corresponding sum for Q for all k.
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REFERENCES
| R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.2(f).
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Let n=binomial(m+1, 2)+r, 0<=r<=m; then a(n) = (1/3)*m*(m^2+3*r-1).
G.f.: (psi(x)-1)*x/(1-x)^2 where psi() is a Ramanujan theta function. - Michael Somos Mar 06 2006
a(n) = sum_(k=0..n) A003056(k) - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 10 2007
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EXAMPLE
| a(6)=8; one longest chain consists of these 9 partitions: 6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1. Others are obtained by changing 3+3 to 4+1+1 or 2+2+2 to 3+1+1+1.
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PROG
| (PARI) {a(n)=local(x); if(n<0, 0, x=(sqrtint(1+8*n)-1)\2; x*(x^2-1+3*(n-x*(x+1)/2))/3)} /* Michael Somos Mar 06 2006 */
(Haskell)
a006463 n = a006463_list !! n
a006463_list = 0 : scanl1 (+) a003056_list
-- Reinhard Zumkeller, Dec 17 2011
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CROSSREFS
| Cf. A076269.
Cf. A003056.
Cf. A010054, A060432.
Sequence in context: A134421 A092777 A186380 * A060655 A117490 A032514
Adjacent sequences: A006460 A006461 A006462 * A006464 A006465 A006466
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 09 2002
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