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A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

%I A277100 #13 Dec 11 2016 12:31:46
%S A277100 1,1,1,1,2,1,4,1,5,2,7,4,10,5,15,6,1,21,8,1,28,13,1,37,18,1,50,25,2,
%T A277100 67,31,3,88,42,5,115,55,6,150,73,8,193,93,11,248,122,15,317,154,19,
%U A277100 402,200,24,1,508,253,30,1,640,320,41,1,802,399,53,1,1002,503,69,1
%N A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).
%C A277100 Sum of entries in row n is A000041(n) (the partition numbers).
%C A277100 T(n,0) = A277102(n).
%C A277100 Sum(k*T(n,k), k>=0) = A277101(n).
%H A277100 Alois P. Heinz, <a href="/A277100/b277100.txt">Rows n = 0..1000, flattened</a>
%F A277100 G.f.:  G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^(i+1))).
%e A277100 The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
%e A277100 T(5,1) = 2, counting [1,1,1,2] and [2,3].
%e A277100 T(8,2) = 1, counting [2,3,3].
%e A277100 Triangle starts:
%e A277100 1;
%e A277100 1;
%e A277100 1, 1;
%e A277100 2, 1;
%e A277100 4, 1;
%e A277100 5, 2;
%e A277100 7, 4;
%e A277100 ...
%p A277100 g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): 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 A277100 # second Maple program:
%p A277100 b:= proc(n, i) option remember; expand(
%p A277100       `if`(n=0, 1, `if`(i<1, 0, add(
%p A277100       `if`(i-1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
%p A277100     end:
%p A277100 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A277100 seq(T(n), n=0..30);  # _Alois P. Heinz_, Oct 10 2016
%t A277100 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_, Dec 08 2016 after _Alois P. Heinz_ *)
%Y A277100 Cf. A000041, A116608, A276427, A276428, A276429, A276433, A276434, A277099, A277101, A277102.
%K A277100 nonn,tabf
%O A277100 0,5
%A A277100 _Emeric Deutsch_, Oct 10 2016