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A276433 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).

%I A276433 #17 Nov 28 2016 09:06:46
%S A276433 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 A276433 104,28,3,137,37,2,177,50,4,229,64,4,295,82,8,377,105,8,477,139,10,1,
%U A276433 605,174,13,761,220,21,956,275,24,1193,350,31,1
%N A276433 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 A276433 Sum of entries in row n is A000041(n) (the partition numbers).
%C A276433 T(n,0) = A277099(n).
%C A276433 Sum(k*T(n,k), k>=0) = A276434(n).
%H A276433 Alois P. Heinz, <a href="/A276433/b276433.txt">Rows n = 0..1000, flattened</a>
%F A276433 G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).
%e A276433 The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
%e A276433 T(5,1) = 1, counting [1,1,3].
%e A276433 T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
%e A276433 T(8,2) = 1, counting [1,1,2,2,2].
%e A276433 Triangle starts:
%e A276433 1;
%e A276433 1;
%e A276433 1,1;
%e A276433 3;
%e A276433 4,1;
%e A276433 6,1;
%e A276433 8,3.
%p A276433 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 A276433 # second Maple program:
%p A276433 b:= proc(n, i) option remember; expand(
%p A276433       `if`(n=0, 1, `if`(i<1, 0, add(
%p A276433       `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
%p A276433     end:
%p A276433 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A276433 seq(T(n), n=0..30);  # _Alois P. Heinz_, Sep 30 2016
%t A276433 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 A276433 Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.
%K A276433 nonn,tabf
%O A276433 0,5
%A A276433 _Emeric Deutsch_, Sep 30 2016