A117192 Number of partitions of n into distinct parts with an even rank.
1, 0, 2, 0, 2, 2, 3, 2, 4, 6, 6, 7, 8, 12, 14, 16, 18, 24, 26, 32, 38, 46, 52, 60, 70, 83, 96, 112, 128, 148, 170, 194, 224, 256, 293, 334, 380, 432, 492, 556, 630, 712, 804, 908, 1026, 1152, 1296, 1454, 1632, 1828, 2048, 2292, 2560, 2858, 3190, 3554, 3959, 4404
Offset: 1
Keywords
Programs
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Mathematica
a[n_] := Count[IntegerPartitions[n], q_ /; EvenQ[First[q] - Length[q]] && Length[q] == Length[Union[q]]]; Array[a, 60] (* Jean-François Alcover, Oct 06 2021 *)
Formula
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
Offset: 1
Comments
Examples
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.Links
Crossrefs
Programs
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 2011Mathematica
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 *)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 */Formula
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