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 A116595 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts that appear exactly once (n>=0, k>=0). 5
 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 4, 4, 2, 1, 2, 8, 4, 1, 6, 8, 6, 2, 5, 12, 10, 3, 9, 16, 12, 4, 1, 7, 23, 19, 6, 1, 16, 24, 25, 10, 2, 11, 40, 33, 14, 3, 22, 45, 41, 22, 5, 20, 59, 63, 27, 6, 1, 33, 72, 73, 42, 10, 1, 28, 99, 101, 53, 14, 2, 51, 108, 127, 75, 21, 3, 42, 153, 167 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Row n has 1+floor([sqrt(1+8n)-1]/2) terms. Row sums yield the partition numbers (A000041). T(n,0)=A007690(n). Sum(k*T(n,k),k>=0)=A024786(n+1). LINKS Alois P. Heinz, Rows n = 0..500, flattened FORMULA G.f.: product(1+tx^j+x^(2j)/(1-x^j), j=1..infinity). More generally, g.f. for the number of partitions of n having exactly k parts that appear exactly m times is product((t-1)*x^(m*j)+1/(1-x^j), j=1..infinity). - Vladeta Jovovic, Feb 21 2006 EXAMPLE T(7,2) = 4 because we have [6,1], [5,2], [4,3], [3,2,1,1]. Triangle starts: 1; 0, 1; 1, 1; 1, 1,  1; 2, 2,  1; 1, 4,  2; 4, 4,  2,  1; 2, 8,  4,  1; 6, 8,  6,  2; 5, 12, 10, 3; 9, 16, 12, 4, 1; MAPLE g:=product(1+t*x^j+x^(2*j)/(1-x^j), j=1..40): gser:=simplify(series(g, x=0, 23)): P[0]:=1: for n from 1 to 21 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 21 do seq(coeff(P[n], t, j), j=0..floor((sqrt(1+8*n)-1)/2)) od; # yields sequence in triangular form # second Maple program: b:= proc(n, i) option remember; local j; if n=0 then 1       elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)       ->x+y, %, [`if`(j=1, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi     end: T:= n-> b(n, n): seq(T(n), n=0..30);  # Alois P. Heinz, Nov 07 2012 MATHEMATICA b[n_, i_] := b[n, i] = Module[{j, pc}, If[n == 0, pc = {1}, If[i<1, pc = {0}, pc = {}; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, If[j == 1, {0}, {}] ~Join~ b[n-i*j, i-1]}]]; pc]]]; T[n_] := b[n, n]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Jan 31 2014, after Alois P. Heinz *) CROSSREFS Cf. A000041, A007690, A024786. Sequence in context: A217209 A233307 A202735 * A128315 A198329 A123566 Adjacent sequences:  A116592 A116593 A116594 * A116596 A116597 A116598 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Feb 18 2006 STATUS approved

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Last modified September 20 13:36 EDT 2020. Contains 337264 sequences. (Running on oeis4.)