cp's OEIS Frontend

A161224 Triangular table a(n,m) that counts the occurrences of m in all partitions of 2n in exactly n parts.

%I A161224 #24 Feb 16 2025 08:33:10
%S A161224 0,0,1,1,2,1,3,4,1,1,8,7,3,1,1,15,12,4,2,1,1,31,19,8,4,2,1,1,51,30,11,
%T A161224 6,3,2,1,1,90,45,19,9,6,3,2,1,1,142,67,26,15,8,5,3,2,1,1,228,97,41,21,
%U A161224 13,8,5,3,2,1,1,341,139,56,31,18,12,7,5,3,2,1,1,525,195,83,45,28,17,12,7,5,3,2,1,1
%N A161224 Triangular table a(n,m) that counts the occurrences of m in all partitions of 2n in exactly n parts.
%C A161224 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).
%C A161224 It appears the row n lists A196087(n) together with the row n of triangle A066633. - _Omar E. Pol_, Feb 26 2012
%H A161224 Alois P. Heinz, <a href="/A161224/b161224.txt">Rows n = 0..140, flattened</a>
%H A161224 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EldersTheorem.html">Elder's Theorem</a>
%e A161224 Table starts:
%e A161224 0;
%e A161224 0, 1;
%e A161224 1, 2, 1;
%e A161224 3, 4, 1, 1;
%e A161224 8, 7, 3, 1, 1;
%e A161224 since the strict partitions of
%e A161224 (2 in 1 part) is {2} with 0 "1" and 1 "2"
%e A161224 (4 in 2 parts) is {2,2}, {3,1} with1 "1", 2 "2" and 1 "3"
%e A161224 (6 in 3 parts) is {2,2,2}, {3,2,1}, {4,1,1} with 3 "1", 4 "2", 1 "3" and 1 "4"
%p A161224 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A161224       `if`(i=1, `if`(t=n, 1+t*x, 0), expand(add((p->p+coeff(
%p A161224       p, x, 0)*j*x^i)(b(n-i*j, i-1, t-j)), j=0..min(t, n/i)))))
%p A161224     end:
%p A161224 a:= n->(p->seq(coeff(p, x, i), i=1..n+1))(b(2*n$2, n)):
%p A161224 seq(a(n), n=0..12);  # _Alois P. Heinz_, Feb 11 2014
%t A161224 <<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}]
%t A161224 (* second program: *)
%t A161224 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]}]]]];
%t A161224 a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n+1}]][b[2n, 2n, n] ];
%t A161224 Table[a[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 24 2016, after _Alois P. Heinz_ *)
%Y A161224 Cf. A066186, A066633.
%K A161224 nonn,tabl
%O A161224 0,5
%A A161224 _Wouter Meeussen_, Jun 06 2009
%E A161224 Row 0 inserted and tabf changed to tabl by _Alois P. Heinz_, Feb 11 2014