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A293136 Irregular triangle T(n,k) read by rows: T(n,k) is the number of strongly unimodal compositions of n (A059618) into k parts.

%I A293136 #22 Jan 29 2024 11:20:37
%S A293136 1,0,1,0,1,0,1,2,0,1,2,1,0,1,4,1,0,1,4,5,0,1,6,6,2,0,1,6,10,4,0,1,8,
%T A293136 14,6,1,0,1,8,19,14,1,0,1,10,23,20,5,0,1,10,31,30,10,0,1,12,36,42,18,
%U A293136 2,0,1,12,44,60,27,4,0,1,14,52,76,48,8,0,1,14,61,102,68,16,1,0,1,16,69,126,101,30,1,0,1,16,81,160,138,50,5,0
%N A293136 Irregular triangle T(n,k) read by rows: T(n,k) is the number of strongly unimodal compositions of n (A059618) into k parts.
%C A293136 Conjecture: index k of last nonzero entry in row n of is A293137(n).
%H A293136 Joerg Arndt, <a href="/A293136/b293136.txt">Table of n, a(n) for n = 0..1793</a> (rows 0...125)
%F A293136 G.f.: 1 + Sum_{n>=1} t*x^n * ( Product_{k=1..n-1} 1 + t*x^k )^2.
%e A293136 Triangle starts:
%e A293136 00:  [1]
%e A293136 01:  [0, 1]
%e A293136 02:  [0, 1]
%e A293136 03:  [0, 1, 2]
%e A293136 04:  [0, 1, 2, 1]
%e A293136 05:  [0, 1, 4, 1]
%e A293136 06:  [0, 1, 4, 5]
%e A293136 07:  [0, 1, 6, 6, 2]
%e A293136 08:  [0, 1, 6, 10, 4]
%e A293136 09:  [0, 1, 8, 14, 6, 1]
%e A293136 10:  [0, 1, 8, 19, 14, 1]
%e A293136 11:  [0, 1, 10, 23, 20, 5]
%e A293136 12:  [0, 1, 10, 31, 30, 10]
%e A293136 13:  [0, 1, 12, 36, 42, 18, 2]
%e A293136 14:  [0, 1, 12, 44, 60, 27, 4]
%e A293136 15:  [0, 1, 14, 52, 76, 48, 8]
%e A293136 16:  [0, 1, 14, 61, 102, 68, 16, 1]
%e A293136 17:  [0, 1, 16, 69, 126, 101, 30, 1]
%e A293136 18:  [0, 1, 16, 81, 160, 138, 50, 5]
%e A293136 19:  [0, 1, 18, 90, 194, 191, 80, 10]
%e A293136 20:  [0, 1, 18, 102, 238, 252, 118, 22]
%e A293136 ...
%e A293136 Row n=7 is [0, 1, 6, 6, 2] because in the 15 partitions of 7 there is 0 into zero parts, 1 into one part, 6 into two parts, 6 into three parts, and 2 into four parts:
%e A293136 [ 1]   [ 1 2 3 1 ]
%e A293136 [ 2]   [ 1 2 4 ]
%e A293136 [ 3]   [ 1 3 2 1 ]
%e A293136 [ 4]   [ 1 4 2 ]
%e A293136 [ 5]   [ 1 5 1 ]
%e A293136 [ 6]   [ 1 6 ]
%e A293136 [ 7]   [ 2 3 2 ]
%e A293136 [ 8]   [ 2 4 1 ]
%e A293136 [ 9]   [ 2 5 ]
%e A293136 [10]   [ 3 4 ]
%e A293136 [11]   [ 4 2 1 ]
%e A293136 [12]   [ 4 3 ]
%e A293136 [13]   [ 5 2 ]
%e A293136 [14]   [ 6 1 ]
%e A293136 [15]   [ 7 ]
%o A293136 (PARI) N=25;  x='x+O('x^N);
%o A293136 T=Vec(1 + sum(n=1, N, t*x^(n) * prod(k=1, n-1, 1+t*x^k)^2));
%o A293136 for(r=1,#T, print(Vecrev(T[r])) );  \\ as triangle
%Y A293136 Cf. A059618 (row sums), A293137.
%Y A293136 Cf. A072704 (same for weakly unimodal compositions).
%K A293136 nonn,tabf
%O A293136 0,8
%A A293136 _Joerg Arndt_, Oct 01 2017