A268190 Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.
1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 7, 5, 2, 2, 1, 1, 3, 11, 5, 5, 2, 2, 1, 1, 4, 13, 10, 5, 4, 2, 2, 1, 1, 2, 20, 11, 8, 5, 4, 2, 2, 1, 1, 6, 23, 16, 10, 8, 4, 4, 2, 2, 1, 1, 2, 33, 20, 15, 9, 8, 4, 4
Offset: 1
Examples
T(5,0)=2 because we have [5] and [1,1,1,1,1]; T(5,1)=3 because we have [3,2], [2,2,1], and [2,1,1,1]; T(5,2)=1 because we have [3,1,1]; T(5,3)=1 because we have [4,1]. Triangle starts: 1; 2; 2,1; 3,1,1; 2,3,1,1; 4,3,2,1,1;
Links
- Alois P. Heinz, Rows n = 1..200, flattened
Programs
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Maple
G := add(x^k/(1-x^k), k = 1 .. 80)+ add(add(t^(i-j)*x^(i+j)/((1-x^i)*mul(1-x^k,k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: 1; for n from 2 to 25 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, l, i) option remember; `if`(irem(n, i)=0, x^ `if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j, `if`(j=0, l, i), i+1), j=0..(n-1)/i)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)): seq(T(n), n=1..30); # Alois P. Heinz, Feb 11 2016 -
Mathematica
b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, x^If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
Formula
G.f.: G(t,x) = Sum_{k>0} (x^k/(1-x^k)) + Sum_{k>1} (Sum_{j=1..i-1} t^{i-j}*x^{i+j}/((1 - x^i)*Product_{k=1..j} (1 - x^k))).
Comments