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Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.
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%I #26 Jan 06 2015 12:11:47

%S 1,1,0,2,0,0,2,1,0,0,3,1,1,0,0,2,3,1,1,0,0,4,3,2,1,1,0,0,2,6,3,2,1,1,

%T 0,0,4,6,6,2,2,1,1,0,0,3,10,6,5,2,2,1,1,0,0,4,11,11,6,4,2,2,1,1,0,0,2,

%U 16,13,10,5,4,2,2,1,1,0,0,6,17,19,12,9,4,4,2,2,1,1,0,0,2,24,24,18,11,8,4,4,2,2,1,1,0,0

%N Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.

%C Reversed rows and also the columns converge to A002865 (setting A002865(0)=0).

%C Column k=0 is A000005 (n>=1), column k=1 is A237665.

%C Row sums are A000041.

%C Sum_{i=0..k} T(n,i) for k=0-9 gives: A000005, A034296, A224956, A238863, A238864, A238865, A238866, A238867, A238868, A238869.

%H Joerg Arndt and Alois P. Heinz, <a href="/A238353/b238353.txt">Rows 0..140, flattened</a>

%F G.f. for column k>=1: sum(j>=1, q^j/(1-q^j) * (prod(i=1..j-1, (1-q^((k+1)*i))/(1-q^i) ) - prod(i=1..j-1, (1-q^(k*i))/(1-q^i) ) ) ), see the comment about the g.f. in A238863.

%e Triangle starts:

%e 00: 1;

%e 01: 1, 0;

%e 02: 2, 0, 0;

%e 03: 2, 1, 0, 0;

%e 04: 3, 1, 1, 0, 0;

%e 05: 2, 3, 1, 1, 0, 0;

%e 06: 4, 3, 2, 1, 1, 0, 0;

%e 07: 2, 6, 3, 2, 1, 1, 0, 0;

%e 08: 4, 6, 6, 2, 2, 1, 1, 0, 0;

%e 09: 3, 10, 6, 5, 2, 2, 1, 1, 0, 0;

%e 10: 4, 11, 11, 6, 4, 2, 2, 1, 1, 0, 0;

%e 11: 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 0, 0;

%e 12: 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 0, 0;

%e 13: 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1, 0, 0;

%e 14: 4, 27, 34, 22, 17, 10, 7, 4, 4, 2, 2, 1, 1, 0, 0;

%e 15: 4, 35, 39, 33, 20, 15, 9, 7, 4, 4, 2, 2, 1, 1, 0, 0;

%e ...

%e The 7 partitions of 5 and their maximal ascents are:

%e 1: [ 1 1 1 1 1 ] 0

%e 2: [ 1 1 1 2 ] 1

%e 3: [ 1 1 3 ] 2

%e 4: [ 1 2 2 ] 1

%e 5: [ 1 4 ] 3

%e 6: [ 2 3 ] 1

%e 7: [ 5 ] 0

%e There are 2 rows with 0 ascents, 3 with 1 ascent, 1 for ascents 2 and 3, giving row 5 of the triangle.

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

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

%p `if`(t=0 or t-i=0, p, add(coeff(p, x, j)*x^

%p max(j, t-i), j=0..degree(p))))(b(n-i, i, i)))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, k), k=0..n))(b(n$2, 0)):

%p seq(T(n), n=0..15);

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, Function[{p}, If[t == 0 || t-i == 0, p, Sum[Coefficient[p, x, j]*x^ Max[j, t-i], {j, 0, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, k], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Jan 06 2015, translated from Maple *)

%Y Cf. A238354 (partitions by minimal ascent).

%K nonn,tabl

%O 0,4

%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014