A213074 Irregular triangle read by rows: coefficients c(n,k) (n>=2, 0<=k<= floor((n-2)/2)) in formula for number of restricted partitions.

1, 1, 1, 2, 1, 3, 1, 7, 8, 1, 10, 14, 1, 17, 50, 36, 1, 24, 89, 78, 1, 36, 207, 368, 200, 1, 49, 340, 701, 431, 1, 70, 685, 2190, 2756, 1188, 1, 93, 1075, 3935, 5564, 2658

Offset

2,4

Comments

Let T^(n)_m denote the number of partitions of mn that can be obtained by adding together m (not necessarily distinct) partitions of n (see A213086). For T^(n)_2, T^(n)_3, T^(n)_4, T^(n)_5 see A002219 through A002222.

Metropolis and Stein show that T^(n)_m is given by the formula

T^(n)_m = Sum_{k=0..n-g-1} binomial(m+g,g+k) c(n,k), where g = floor((n+1)/2).

Links

Table of n, a(n) for n=2..43.

N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

Example

Triangle c(n,k) begins:
n\k
-  0    1    2    3    4    5 ...
---------------------------------
2  1
3  1
4  1    2
5  1    3
6  1    7    8
7  1   10   14
8  1   17   50   36
9  1   24   89   78
10 1   36  207  368  200
11 1   49  340  701  431
12 1   70  685 2190 2756 1188
13 1   93 1075 3935 5564 2658
...

Maple

with(combinat):
h:= proc(n, m) option remember;
      `if`(m>1, map(x-> map(y-> sort([x[], y[]]), h(n, 1))[],
       h(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
       {partition(n)[]}), {[]}))
    end:
T:= proc(n) local i, g, t;
      g:= floor((n+1)/2);
      subs(solve({seq(nops(h(n, t))=add(c||i *binomial(t+g, g+i),
      i=0..n-g-1), t=1..n-g)}, {seq(c||i, i=0..n-g-1)}),
      [seq(c||i, i=0..n-g-1)])[]
    end:
seq(T(n), n=2..10);  # Alois P. Heinz, Jul 11 2012

Mathematica

nmax = 13; mmax = 5;
T[n_, m_] := T[n, m] = Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ip]; i[0] = 1; Table[ Join[ Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k - 1], lg}, {k, 1, m}]]] // Flatten[#, m - 1] & // Union // Length]; T[_, 0] = 1;
U[n_, m_] := With[{g = Floor[(n + 1)/2]}, If[n == 1, 1, Sum[Binomial[m + g, g + k] c[n, k], {k, 0, n - g - 1}]]];
Do[TT = Table[T[n , m] - U[n , m], {n, 1, nmax}, {m, 0, mm}] // Flatten; c[_, 0] = 1; sol = Solve[Thread[TT == 0]][[1]]; cc = Table[c[n, k], {n, 2, nmax}, {k, 0, Floor[(n - 2)/2]}] /. sol // Flatten; Print[cc], {mm, 2, mmax}];
cc (* Jean-François Alcover, May 25 2016 *)

Crossrefs

Cf. A000041, A002219, A002220, A002221, A002222, A213075, A213076, A213086.

Sequence in context: A370906 A165401 A319916 * A140966 A058036 A373986

Adjacent sequences: A213071 A213072 A213073 * A213075 A213076 A213077

Keyword

nonn,tabf,more

Author

N. J. A. Sloane, Jun 04 2012

Extensions

12 more terms (rows 12-13) from Alois P. Heinz, Jul 11 2012

Status

approved