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.
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, 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, 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
Offset: 0
Examples
Triangle starts: 00: [1] 01: [0, 1] 02: [0, 1] 03: [0, 1, 2] 04: [0, 1, 2, 1] 05: [0, 1, 4, 1] 06: [0, 1, 4, 5] 07: [0, 1, 6, 6, 2] 08: [0, 1, 6, 10, 4] 09: [0, 1, 8, 14, 6, 1] 10: [0, 1, 8, 19, 14, 1] 11: [0, 1, 10, 23, 20, 5] 12: [0, 1, 10, 31, 30, 10] 13: [0, 1, 12, 36, 42, 18, 2] 14: [0, 1, 12, 44, 60, 27, 4] 15: [0, 1, 14, 52, 76, 48, 8] 16: [0, 1, 14, 61, 102, 68, 16, 1] 17: [0, 1, 16, 69, 126, 101, 30, 1] 18: [0, 1, 16, 81, 160, 138, 50, 5] 19: [0, 1, 18, 90, 194, 191, 80, 10] 20: [0, 1, 18, 102, 238, 252, 118, 22] ... 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: [ 1] [ 1 2 3 1 ] [ 2] [ 1 2 4 ] [ 3] [ 1 3 2 1 ] [ 4] [ 1 4 2 ] [ 5] [ 1 5 1 ] [ 6] [ 1 6 ] [ 7] [ 2 3 2 ] [ 8] [ 2 4 1 ] [ 9] [ 2 5 ] [10] [ 3 4 ] [11] [ 4 2 1 ] [12] [ 4 3 ] [13] [ 5 2 ] [14] [ 6 1 ] [15] [ 7 ]
Links
- Joerg Arndt, Table of n, a(n) for n = 0..1793 (rows 0...125)
Programs
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PARI
N=25; x='x+O('x^N); T=Vec(1 + sum(n=1, N, t*x^(n) * prod(k=1, n-1, 1+t*x^k)^2)); for(r=1,#T, print(Vecrev(T[r])) ); \\ as triangle
Formula
G.f.: 1 + Sum_{n>=1} t*x^n * ( Product_{k=1..n-1} 1 + t*x^k )^2.
Comments