A118399 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).

1, 1, 2, 2, 3, 5, 6, 8, 11, 15, 18, 24, 29, 37, 47, 57, 69, 86, 103, 125, 154, 183, 220, 264, 316, 375, 450, 533, 631, 747, 882, 1035, 1222, 1428, 1674, 1959, 2282, 2653, 3088, 3578, 4142, 4790, 5525, 6363, 7330, 8410, 9644, 11050, 12633, 14424, 16459, 18743

Offset: 1

Author

Paul D. Hanna, May 07 2006

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A008289.

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= proc(n) option remember; local l; l:= b(n, n);
      `if`(n=1, 1, add(l[i+1]*a(i), i=1..nops(l)-1))
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Nov 18 2012

Mathematica

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 *)

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))))}