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 A117195 Triangle read by rows: T(n,k) = number of partitions into distinct parts having rank k, 0<=k
 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,40 COMMENTS T(n,0) = A010054(n), T(n,1) = 1-A010054(n) for n>1; A000009(n) = Sum(T(n,k): 0<=k=1, q^(n*(n+1)/2) / prod(k=1..n, 1-z*q^k) ), see Monks reference. [Joerg Arndt, Oct 07 2012] EXAMPLE Triangle starts: [ 1]   1, [ 2]   0, 1, [ 3]   1, 0, 1, [ 4]   0, 1, 0, 1, [ 5]   0, 1, 1, 0, 1, [ 6]   1, 0, 1, 1, 0, 1, [ 7]   0, 1, 1, 1, 1, 0, 1, [ 8]   0, 1, 1, 1, 1, 1, 0, 1, [ 9]   0, 1, 1, 2, 1, 1, 1, 0, 1, [10]   1, 0, 2, 1, 2, 1, 1, 1, 0, 1, [11]   0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, [12]   0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, [13]   0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1, [14]   0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1, ... T(12,0) = #{} = 0, T(12,1) = #{5+4+2+1} = 1, T(12,2) = #{6+3+2+1, 5+4+3} = 2, T(12,3) = #{6+5+1, 6+4+2} = 2, T(12,4) = #{7+4+1, 7+3+2} = 2, T(12,5) = #{8+3+1, 7+5} = 2, T(12,6) = #{9+2+1, 8+4} = 2, T(12,7) = #{9+3} = 1, T(12,8) = #{10+2} = 1, T(12,9) = #{11+1} = 1, T(12,10) = #{} = 0, T(12,11) = #{12} = 1. MAPLE b:= proc(n, i, k) option remember;       if n<0 or k<0 then []     elif n=0 then [0\$k, 1]     elif i<1 then []     else zip ((x, y)-> x+y, b(n, i-1, k), b(n-i, i-1, k-1), 0)       fi     end: T:= proc(n) local j, r; r:= [];       for j from 0 to n do         r:= zip ((x, y)-> x+y, r, b(n-j, j-1, j-1), 0)       od; r[]     end: seq (T(n), n=1..20);  # Alois P. Heinz, Aug 29 2011 MATHEMATICA b[n_, i_, k_] := b[n, i, k] = Which[n<0 || k<0, {}, n == 0, Append[Array[0&, k], 1], i<1, {}, True, Plus @@ PadRight[{b[n, i-1, k], b[n-i, i-1, k-1]}]]; T[n_] := Module[{j, r}, r = {}; For[j = 0, j <= n, j++, r = Plus @@ PadRight[{r, b[n-j, j-1, j-1]}]]; r]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *) PROG (PARI) N=33;  L=1+2*ceil(sqrtint(N)); q='q+O(q^N); gf=sum(n=1, L, q^(n*(n+1)/2) / prod(k=1, n, 1-z*q^k) ); v=Vec(gf); { for (n=1, #v,  /* print triangle: */     p = Pol(v[n], 'z) + 'c0;     p = polrecip(p);     rw = Vec(p);  rw[1] -= 'c0;     print1("[", n, "]   " );     print( rw ); ); } /* Joerg Arndt, Oct 07 2012 */ CROSSREFS Cf. A063995, A105806. Sequence in context: A260413 A053252 A261029 * A156606 A324606 A194087 Adjacent sequences:  A117192 A117193 A117194 * A117196 A117197 A117198 KEYWORD nonn,tabl AUTHOR Reinhard Zumkeller, Mar 03 2006 STATUS approved

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Last modified April 13 00:24 EDT 2021. Contains 342934 sequences. (Running on oeis4.)