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A006463 Convolve natural numbers with characteristic function of triangular numbers. 3
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; text; internal format)
OFFSET

0,4

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (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.

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.2(f).

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

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-1) A003056(k). - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 10 2007

a(n+1) - 2*a(n) + a(n-1) = A010054(n) if n>0. - Michael Somos, May 07 2016

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.

G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 17*x^9 + ...

MATHEMATICA

a[n_] := (x = Quotient[ Sqrt[1+8*n]-1, 2]; x*(x^2-1+3*(n-x*(x+1)/2))/3); Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Apr 11 2013, after Michael Somos *)

t = {0}; Do[Do[AppendTo[t, t[[-1]]+n], {k, 0, n}], {n, 0, 11}]; t (* Jean-François Alcover, May 10 2016, after Vladimir Joseph Stephan Orlovsky *)

PROG

(PARI) {a(n) = my(x); if( n<0, 0, x = (sqrtint(8*n + 1) - 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

CROSSREFS

Cf. A076269.

Cf. A003056.

Cf. A010054, A060432, A003056.

Sequence in context: A134421 A092777 A186380 * A237886 A212555 A060655

Adjacent sequences:  A006460 A006461 A006462 * A006464 A006465 A006466

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Dean Hickerson, Nov 09 2002

STATUS

approved

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Last modified May 22 13:18 EDT 2017. Contains 286872 sequences.