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%I #61 Jan 15 2024 19:47:49
%S 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,
%T 94,100,106,112,119,126,133,140,147,154,161,168,176,184,192,200,208,
%U 216,224,232,240,249,258,267,276,285,294,303,312,321,330,340,350,360
%N Convolve natural numbers with characteristic function of triangular numbers.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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.
%C If C_n(q, t) are the (q, t)-Catalan polynomials, then p_n(x) := C_n(x, x) is a polynomial in x such that a(n) is the degree of the lowest degree term. The sequence of polynomials p_n(x) = 1, 1, 2*x, x^2 + 4*x^3, 3*x^4 + 4*x^5 + 7*x^6 + ... while the coefficient of the lowest degree term is A074909(n). - _Michael Somos_, Jan 09 2019
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.2(f).
%H Reinhard Zumkeller, <a href="/A006463/b006463.txt">Table of n, a(n) for n = 0..10000</a>
%H Jeffrey Shallit, <a href="/A006463/a006463.pdf">Letter to N. J. A. Sloane with attachment, Aug. 1979</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Let n=binomial(m+1, 2)+r, 0<=r<=m; then a(n) = (1/3)*m*(m^2+3*r-1).
%F G.f.: (psi(x) - 1) * x / (1 - x)^2 where psi() is a Ramanujan theta function. - _Michael Somos_, Mar 06 2006
%F a(n) = Sum_(k=0..n-1) A003056(k). - _Daniele Parisse_, Jul 10 2007
%F a(n+1) - 2*a(n) + a(n-1) = A010054(n) if n>0. - _Michael Somos_, May 07 2016
%e 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.
%e 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 + ...
%t 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 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_ *)
%t Join[{0},Table[ListConvolve[Range[x],Table[If[OddQ[Sqrt[8n+1]],1,0],{n,x}]],{x,0,60}]//Flatten] (* _Harvey P. Dale_, Jan 14 2019 *)
%o (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 */
%o (Haskell)
%o a006463 n = a006463_list !! n
%o a006463_list = 0 : scanl1 (+) a003056_list
%o -- _Reinhard Zumkeller_, Dec 17 2011
%Y Cf. A076269, A074909, A010054, A060432.
%Y 0 together with the partial sums of A003056.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E Edited by _Dean Hickerson_, Nov 09 2002
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