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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 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

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