A264402 - OEIS

%I #17 Jan 15 2016 15:42:54

%S 1,2,2,1,3,2,2,4,1,4,5,2,2,8,4,1,4,9,7,2,3,12,10,4,1,4,14,15,7,2,2,17,

%T 20,12,4,1,6,18,27,17,7,2,2,23,33,26,12,4,1,4,24,44,35,19,7,2,4,27,51,

%U 49,28,12,4,1,5,30,64,63,41,19,7,2,2

%N Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

%C T(n,k) = number of partitions of n in which the 2nd largest part is k (0 if all parts are equal). Example: T(7,2) = 4 because we have [3,2,1,1], [3,2,2], [4,2,1], and [5,2].

%C The fact that the above two statistics (in Name and in 1st Comment) have the same distribution follows at once by conjugation. - _Emeric Deutsch_, Dec 11 2015

%C Row sums yield the partition numbers (A000041).

%C T(n,0) = A000005(n) = number of divisors of n.

%C Sum_{k>=0} k*T(n,k) = A182984(n).

%H Alois P. Heinz, <a href="/A264402/b264402.txt">Rows n = 1..350, flattened</a>

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

%e T(7,2) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,2,1].

%e Triangle starts:

%e 1;

%e 2;

%e 2,1;

%e 3,2;

%e 2,4,1;

%e 4,5,2;

%e 2,8,4,1;

%p g := sum(x^i/((1-x^i)*(product(1-t*x^j, j = i+1 .. 100))), i = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n to 27 do P[n] := sort(coeff(gser, x, n)) end do: for n to 27 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p       `if`(i<1, 0, b(n, i-1) +add((p->[0, p[1]+

%p        expand(p[2]*x^j)])(b(n-i*j, i-1)) , j=1..n/i)))

%p     end:

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

%p seq(T(n), n=1..20);  # _Alois P. Heinz_, Nov 29 2015

%t b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[p, {0, p[[1]] + Expand[p[[2]]*x^j]}][b[n-i*j, i-1]], {j, 1, n/i} ]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n][[2]]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Jan 15 2016, after _Alois P. Heinz_ *)

%Y Cf. A000005, A000041, A116685, A182984, A002541.

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Nov 21 2015