cp's OEIS Frontend

A268190 Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.

%I A268190 #14 Dec 21 2016 08:17:24
%S A268190 1,2,2,1,3,1,1,2,3,1,1,4,3,2,1,1,2,6,3,2,1,1,4,7,5,2,2,1,1,3,11,5,5,2,
%T A268190 2,1,1,4,13,10,5,4,2,2,1,1,2,20,11,8,5,4,2,2,1,1,6,23,16,10,8,4,4,2,2,
%U A268190 1,1,2,33,20,15,9,8,4,4
%N A268190 Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.
%C A268190 Sum of entries in row n is A000041(n) (the partition numbers).
%C A268190 T(n,0) = A000005(n) = number of divisors of n.
%C A268190 T(n,1) = A083751(n+1) for n>=3.
%C A268190 Sum(k*T(n,k),k>=1) = A268191(n).
%C A268190 T(2n,n) = A002865(n) for n>=2. - _Alois P. Heinz_, Feb 11 2016
%H A268190 Alois P. Heinz, <a href="/A268190/b268190.txt">Rows n = 1..200, flattened</a>
%F A268190 G.f.: G(t,x) = Sum_{k>0} (x^k/(1-x^k)) + Sum_{k>1} (Sum_{j=1..i-1} t^{i-j}*x^{i+j}/((1 - x^i)*Product_{k=1..j} (1 - x^k))).
%e A268190 T(5,0)=2 because we have [5] and [1,1,1,1,1]; T(5,1)=3 because we have [3,2], [2,2,1], and [2,1,1,1]; T(5,2)=1 because we have [3,1,1]; T(5,3)=1 because we have [4,1].
%e A268190 Triangle starts:
%e A268190 1;
%e A268190 2;
%e A268190 2,1;
%e A268190 3,1,1;
%e A268190 2,3,1,1;
%e A268190 4,3,2,1,1;
%p A268190 G := add(x^k/(1-x^k), k = 1 .. 80)+ add(add(t^(i-j)*x^(i+j)/((1-x^i)*mul(1-x^k,k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: 1; for n from 2 to 25 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
%p A268190 # second Maple program:
%p A268190 b:= proc(n, l, i) option remember; `if`(irem(n, i)=0, x^
%p A268190       `if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j,
%p A268190       `if`(j=0, l, i), i+1), j=0..(n-1)/i))
%p A268190     end:
%p A268190 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A268190 seq(T(n), n=1..30);  # _Alois P. Heinz_, Feb 11 2016
%t A268190 b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, x^If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 1, 30}] // Flatten (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%Y A268190 Cf. A000005, A000041, A002865, A083751, A268191.
%K A268190 nonn,tabf
%O A268190 1,2
%A A268190 _Emeric Deutsch_, Feb 10 2016