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%I #11 Feb 12 2017 10:11:46
%S 1,1,2,2,3,5,6,8,11,15,18,24,29,37,47,57,69,86,103,125,154,183,220,
%T 264,316,375,450,533,631,747,882,1035,1222,1428,1674,1959,2282,2653,
%U 3088,3578,4142,4790,5525,6363,7330,8410,9644,11050,12633,14424,16459,18743
%N Eigenvector of the triangle of distinct partitions (A008289), so that: a(n) = Sum_{k=1..tri(n)} A008289(n,k)*a(k) for n>=1 with a(1)=1, where tri(n) = floor((sqrt(8*n+1)-1)/2).
%H Alois P. Heinz, <a href="/A118399/b118399.txt">Table of n, a(n) for n = 1..1000</a>
%p b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
%p -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
%p end:
%p a:= proc(n) option remember; local l; l:= b(n, n);
%p `if`(n=1, 1, add(l[i+1]*a(i), i=1..nops(l)-1))
%p end:
%p seq (a(n), n=1..60); # _Alois P. Heinz_, Nov 18 2012
%t zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, zip[b[n, i-1], If[i>n, {}, Join[{0}, b[n-i, i-1]]]]]]; a[n_] := a[n] = ( l = b[n, n]; If[n == 1, 1, Sum[l[[i+1]]*a[i], {i, 1, Length[l]-1}]]); Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Feb 12 2017, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=if(n<0,0,if(n==1,1,sum(k=1,floor((sqrt(8*n+1)-1)/2), a(k)*polcoeff(polcoeff(prod(i=1,n, 1+y*x^i,1+x*O(x^n)),n,x),k,y))))}
%Y Cf. A008289.
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 07 2006
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