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A161224 Triangular table a(n,m) that counts the occurrences of m in all partitions of 2n in exactly n parts. 2
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, May 24 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A066186, A066633.

Sequence in context: A160188 A322081 A279396 * A147567 A247045 A084579

Adjacent sequences:  A161221 A161222 A161223 * A161225 A161226 A161227

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

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)