A276433 - OEIS

%I #17 Nov 28 2016 09:06:46

%S 1,1,1,1,3,4,1,6,1,8,3,12,3,18,3,1,24,6,32,10,45,10,1,59,17,1,79,21,1,

%T 104,28,3,137,37,2,177,50,4,229,64,4,295,82,8,377,105,8,477,139,10,1,

%U 605,174,13,761,220,21,956,275,24,1193,350,31,1

%N Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

%C Sum of entries in row n is A000041(n) (the partition numbers).

%C T(n,0) = A277099(n).

%C Sum(k*T(n,k), k>=0) = A276434(n).

%H Alois P. Heinz, <a href="/A276433/b276433.txt">Rows n = 0..1000, flattened</a>

%F G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).

%e The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.

%e T(5,1) = 1, counting [1,1,3].

%e T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].

%e T(8,2) = 1, counting [1,1,2,2,2].

%e Triangle starts:

%e 1;

%e 1;

%e 1,1;

%e 3;

%e 4,1;

%e 6,1;

%e 8,3.

%p G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n],t,k),k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, i) option remember; expand(

%p       `if`(n=0, 1, `if`(i<1, 0, add(

%p       `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):

%p seq(T(n), n=0..30);  # _Alois P. Heinz_, Sep 30 2016

%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* _Jean-François Alcover_, Nov 28 2016 after _Alois P. Heinz_ *)

%Y Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Sep 30 2016