cp's OEIS Frontend

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).

%I A266477 #39 May 24 2018 18:32:24
%S A266477 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,
%T A266477 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,
%U A266477 1,1,1,1,0,1,1,15,15,9,11,3,6,2,5,3,3,0
%N 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).
%C A266477 Sum of entries in row n = A000041(n) = number of partitions of n.
%C A266477 T(n,1) = A000009(n) = number of partitions of n into distinct parts.
%C A266477 T(n,2) = A090858(n).
%C A266477 T(n,3) = A265251(n).
%C A266477 Smallest row m >= 0 with T(m,n) > 0 is A266325(n).
%C A266477 T(n,A266480(n)) gives A266871(n).
%H A266477 Alois P. Heinz, <a href="/A266477/b266477.txt">Rows n = 0..50, flattened</a>
%F A266477 Sum_{k>=1} k*T(n,k) = A077285(n).
%F A266477 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.
%F A266477 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
%e A266477 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.
%e A266477 Triangle T(n,k) begins:
%e A266477 00 :  1;
%e A266477 01 :  1;
%e A266477 02 :  1,  1;
%e A266477 03 :  2,  0, 1;
%e A266477 04 :  2,  2, 0,  1;
%e A266477 05 :  3,  2, 1,  0, 1;
%e A266477 06 :  4,  2, 2,  2, 0, 1;
%e A266477 07 :  5,  4, 2,  1, 1, 1, 1;
%e A266477 08 :  6,  6, 2,  3, 1, 2, 0, 2;
%e A266477 09 :  8,  7, 4,  4, 1, 2, 1, 0, 2, 1;
%e A266477 10 : 10,  8, 6,  6, 3, 2, 1, 3, 0, 1, 0, 2;
%e A266477 11 : 12, 13, 6,  6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
%e A266477 12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;
%p A266477 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A266477       x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
%p A266477     end:
%p A266477 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
%p A266477 seq(T(n), n=0..16);
%t A266477 Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* _Michael De Vlieger_, Dec 31 2015 *)
%t A266477 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_ *)
%Y A266477 Columns k=1-10 give: A000009, A090858, A265251, A266687, A266688, A266689, A266690, A266691, A266692, A266693.
%Y A266477 Main diagonal gives A266499.
%Y A266477 Row lengths give A266480.
%Y A266477 Cf. A000041, A077285, A266325, A266871.
%K A266477 nonn,tabf
%O A266477 0,5
%A A266477 _Emeric Deutsch_ and _Alois P. Heinz_, Dec 29 2015