A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

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

Offset: 0

Author

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

Keywords

• nonn
• tabf

Comments

Sum of entries in row n = A000041(n) = number of partitions of n.

T(n,1) = A000009(n) = number of partitions of n into distinct parts.

T(n,2) = A090858(n).

T(n,3) = A265251(n).

Smallest row m >= 0 with T(m,n) > 0 is A266325(n).

T(n,A266480(n)) gives A266871(n).

Examples

			Row 4 is [2,2,0,1]. Indeed, the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
Triangle T(n,k) begins:
00 :  1;
01 :  1;
02 :  1,  1;
03 :  2,  0, 1;
04 :  2,  2, 0,  1;
05 :  3,  2, 1,  0, 1;
06 :  4,  2, 2,  2, 0, 1;
07 :  5,  4, 2,  1, 1, 1, 1;
08 :  6,  6, 2,  3, 1, 2, 0, 2;
09 :  8,  7, 4,  4, 1, 2, 1, 0, 2, 1;
10 : 10,  8, 6,  6, 3, 2, 1, 3, 0, 1, 0, 2;
11 : 12, 13, 6,  6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened

Crossrefs

Columns k=1-10 give: A000009, A090858, A265251, A266687, A266688, A266689, A266690, A266691, A266692, A266693.

Main diagonal gives A266499.

Row lengths give A266480.

Cf. A000041, A077285, A266325, A266871.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..16);

Mathematica

Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^Max[p, p*n], Sum[b[n - i*j, i - 1, Max[p, p*j]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p,x]}]][ b[n, n, 1]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Formula

Sum_{k>=1} k*T(n,k) = A077285(n).

G.f. of column p if p is prime: Sum_{k>0} x^(p*k)/(1+x^k) * Product_{i>0} (1+x^i), giving the number of partitions of n such that there is exactly one part which occurs p times, while all other parts occur only once.

If p is prime then column p is asymptotic to 3^(1/4) * c(p) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)), where c(p) = Sum_{j>=0} (-1)^j/(j+p) = (PolyGamma((p+1)/2) - PolyGamma(p/2))/2. - Vaclav Kotesovec, May 24 2018