A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 9, 9, 1, 5, 14, 23, 23, 1, 6, 20, 43, 66, 1, 7, 27, 70, 136, 136, 1, 8, 35, 105, 241, 377, 377, 1, 9, 44, 149, 390, 767, 1144, 1, 10, 54, 203, 593, 1360, 2504, 2504, 1, 11, 65, 268, 861, 2221, 4725, 7229, 7229, 1, 12, 77, 345, 1206

Offset: 0

Author

Gerald McGarvey, Oct 03 2009

Keywords

• nonn
• tabf

Comments

There are f(n) = floor((2*n+3)/3) = A004396(n+1) terms in row n.

Examples

			Triangle begins:
       k=0   1   2    3     4     5     6      7      8
  n=0:   1
  n=1:   1
  n=2:   1,  1
  n=3:   1,  2,  2
  n=4:   1,  3,  5
  n=5:   1,  4,  9,   9
  n=6:   1,  5, 14,  23,   23
  n=7:   1,  6, 20,  43,   66
  n=8:   1,  7, 27,  70,  136,  136
  n=9:   1,  8, 35, 105,  241,  377,  377
  n=10:  1,  9, 44, 149,  390,  767, 1144
  n=11:  1, 10, 54, 203,  593, 1360, 2504,  2504
  n=12:  1, 11, 65, 268,  861, 2221, 4725,  7229,  7229
  n=13:  1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610
  ...
		

Crossrefs

Cf. A004396 (row lengths).

Cf. A060941 (diagonal T(3*n, 2*n)).

Programs

PARI

f(n) = floor((2*(n-1)+3)/3); s=14; M=matrix(s,s); for(n=1,s,M[n,1]=1); for(n=2,s,for(k=2,f(n),M[n,k]=M[n,k-1]+M[n-1,k])); for(n=1,s,for(k=1,f(n),print1(M[n,k],", ")))