A268189 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts larger than the smallest part is k (n>=1, 0<=k<=n-1).

1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 1, 2, 2, 4, 0, 1, 1, 3, 2, 2, 0, 1, 2, 3, 3, 4, 4, 0, 1, 1, 3, 3, 6, 4, 3, 0, 1, 2, 2, 4, 5, 6, 7, 4, 0, 1, 1, 4, 2, 7, 5, 10, 8, 2, 0, 1, 2, 2, 4, 5, 7, 9, 12, 12, 6, 0, 1, 1, 3, 2, 7, 5, 11, 10, 17, 14, 2, 0, 1, 2, 3, 4, 4, 8, 8, 13, 15, 20, 21

Offset

1,2

Comments

Sum of entries in row n is A000041(n).

T(n,0) = A000005(n) = number of divisors of n.

Sum_{k>0} k*T(n,k) = A213359(n).

Links

Alois P. Heinz, Rows n = 1..200, flattened

Formula

G.f.: G(t,x) = Sum_{i>0} x^i/ ((1-x^i)*Product_ {j>i}(1-t^j*x^j)).

Example

T(5,3) = 2 because in the partitions [1,1,3] and [2,3} of 5 the sum of the parts larger than the smallest part is 3.
Triangle starts:
1;
2, 0;
2, 0, 1;
3, 0, 1, 1;
2, 0, 1, 2, 2;
4, 0, 1, 1, 3, 2;
2, 0, 1, 2, 3, 3, 4;
4, 0, 1, 1, 3, 3, 6, 4;
3, 0, 1, 2, 2, 4, 5, 6,  7;
4, 0, 1, 1, 4, 2, 7, 5, 10,  8;
2, 0, 1, 2, 2, 4, 5, 7,  9, 12, 12;
6, 0, 1, 1, 3, 2, 7, 5, 11, 10, 17, 14;

Maple

g := add(x^i/((1-x^i)*mul(1-t^j*x^j, j = i+1 .. 100)), i = 1 .. 100); gser := simplify(series(g, x = 0, 30)); for n to 18 do P[n] := sort(coeff(gser, x, n)) end do; for n to 18 do seq(coeff(P[n], t, j), j = 0 .. n-1) end do
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(irem(n, i)=0, 1, 0)
       +`if`(i>1, add(b(n-i*j, i-1)*x^(i*j), j=0..(n-1)/i), 0))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Feb 04 2016

Mathematica

b[n_, i_] := b[n, i] = Expand[If[Mod[n, i] == 0, 1, 0] + If[i > 1, Sum[b[n - i*j, i - 1]*x^(i*j), {j, 0, (n - 1)/i}], 0]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000005, A000041, A213359, A264402.

Sequence in context: A321299 A105783 A326819 * A265247 A022879 A340524

Adjacent sequences: A268186 A268187 A268188 * A268190 A268191 A268192

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 01 2016

Status

approved