OFFSET
0,5
COMMENTS
Row sums are A066186, or n*p(n) with p(n) = A000041 = the partitions of n. The rows reversed converge to 1,1,2,3,5,7,11,15,... or p(n). The count of partitions of 2n in exactly n parts equals p(n).
It appears the row n lists A196087(n) together with the row n of triangle A066633. - Omar E. Pol, Feb 26 2012
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Elder's Theorem
EXAMPLE
Table starts:
0;
0, 1;
1, 2, 1;
3, 4, 1, 1;
8, 7, 3, 1, 1;
since the strict partitions of
(2 in 1 part) is {2} with 0 "1" and 1 "2"
(4 in 2 parts) is {2,2}, {3,1} with1 "1", 2 "2" and 1 "3"
(6 in 3 parts) is {2,2,2}, {3,2,1}, {4,1,1} with 3 "1", 4 "2", 1 "3" and 1 "4"
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
`if`(i=1, `if`(t=n, 1+t*x, 0), expand(add((p->p+coeff(
p, x, 0)*j*x^i)(b(n-i*j, i-1, t-j)), j=0..min(t, n/i)))))
end:
a:= n->(p->seq(coeff(p, x, i), i=1..n+1))(b(2*n$2, n)):
seq(a(n), n=0..12); # Alois P. Heinz, Feb 11 2014
MATHEMATICA
<<Combinatorica`; partitionexact[n_, m_]:= TransposePartition /@ (Prepend[ #, m]& /@ Partitions[n-m, m]); Table[If[n==0, {0}, CoefficientList[ Apply[ Plus, x^#& /@ partitionexact[2n, n], {0, 1}]/x, x]], {n, 0, 24}]
(* second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i == 1, If[t == n, 1+t*x, 0], Expand[Sum[Function[p, p + Coefficient[p, x, 0]*j*x^i][ b[n-i*j, i-1, t-j]], {j, 0, Min[t, n/i]}]]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n+1}]][b[2n, 2n, n] ];
Table[a[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 24 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Jun 06 2009
EXTENSIONS
Row 0 inserted and tabf changed to tabl by Alois P. Heinz, Feb 11 2014
STATUS
approved