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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. 10
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A370906 A165401 A319916 * A140966 A058036 A373986
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

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Last modified July 23 06:44 EDT 2024. Contains 374544 sequences. (Running on oeis4.)