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A265247 Triangle read by rows: T(n,k) is the number of partitions of n in which the 2nd smallest part is k when the partition has at least 2 distinct parts and 0 otherwise; (n>=1, 0 <= k <= n).

%I A265247 #13 Dec 25 2015 02:42:55
%S A265247 1,2,0,2,0,1,3,0,1,1,2,0,2,2,1,4,0,3,1,2,1,2,0,5,3,2,2,1,4,0,7,4,2,2,
%T A265247 2,1,3,0,11,6,2,3,2,2,1,4,0,15,8,6,1,3,2,2,1,2,0,22,12,6,4,2,3,2,2,1,
%U A265247 6,0,30,15,9,4,3,2,3,2,2,1,2,0,42,22,11,8,2,4,2,3,2,2,1,4,0,56,28,16,10,6,1,4,2,3,2,2,1,4,0,77,38,19,11,7,4,2,4,2,3,2,2,1
%N A265247 Triangle read by rows: T(n,k) is the number of partitions of n in which the 2nd smallest part is k when the partition has at least 2 distinct parts and 0 otherwise; (n>=1, 0 <= k <= n).
%C A265247 Number of entries in row n is n.
%C A265247 Sum of entries in row n is A000041(n) = number of partitions of n.
%C A265247 T(n,0) = A000005(n) = number of divisors of n.
%C A265247 T(n,1) = 0.
%C A265247 T(n,2) = A000041(n-3), i.e., for n>=3 the number of partitions of n having 2 as the 2nd smallest part is equal to the number of partitions of n-3 (follows from a simple bijection: delete a part 2 and a part 1).
%C A265247 Sum_{k>=0} k*T(n,k) = A265248(n).
%F A265247 G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) (1 + Sum_{j>=i+1} t^j*x^j/Product_{k>=j}(1-x^k)).
%e A265247 T(5,3) = 2 because of [3,2] and [3,1,1].
%e A265247 Triangle starts:
%e A265247   1;
%e A265247   2, 0;
%e A265247   2, 0, 1;
%e A265247   3, 0, 1, 1;
%e A265247   2, 0, 2, 2, 1;
%e A265247   4, 0, 3, 1, 2, 1.
%p A265247 g := add(x^i*(1+add(t^j*x^j/(mul(1-x^k, k=j..80)), j=i+1..80))/(1-x^i), i=1..80):
%p A265247 gser := simplify(series(g, x = 0, 25)):
%p A265247 for n to 20 do P[n] := sort(coeff(gser, x, n)) end do:
%p A265247 for n to 20 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
%Y A265247 Cf. A000005, A000041, A265248.
%K A265247 nonn,tabl
%O A265247 1,2
%A A265247 _Emeric Deutsch_, Dec 24 2015