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%I #14 Dec 21 2016 09:38:34
%S 1,2,2,1,4,1,4,3,8,2,1,8,6,1,13,7,2,15,11,4,22,15,4,1,24,24,7,1,37,26,
%T 12,2,40,42,16,3,57,50,22,6,64,72,33,6,1,89,84,46,11,1,98,122,60,15,2,
%U 135,141,82,24,3,149,198,106,32,5,199,231,144,45,8,224,309,187,61,10,1
%N Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.
%C T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
%C Sum of entries in row n is A000041(n) (the partition numbers).
%C T(n,0) = A116931(n).
%C Sum(k*T(n,k), k>=1) = A024786(n-1).
%H Alois P. Heinz, <a href="/A268193/b268193.txt">Rows n = 1..800, flattened</a>
%F G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).
%e T(5,1) = 3 because we have [3,2], [[2,2,1], and [2,1,1,1].
%e T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
%e Triangle starts:
%e 1;
%e 2;
%e 2,1;
%e 4,1;
%e 4,3;
%e 8,2,1;
%e 8,6,1;
%p g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
%p `if`(t and j=1, x, 1), j=0..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
%p seq(T(n), n=1..20); # _Alois P. Heinz_, Feb 13 2016
%t b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%Y Cf. A000041, A024786, A116931.
%K nonn,look,tabf
%O 1,2
%A _Emeric Deutsch_, Feb 13 2016
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