A179748 Triangle T(n,k) read by rows. T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 5, 4, 1, 1, 1, 2, 6, 9, 5, 1, 1, 1, 2, 6, 15, 14, 6, 1, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1

Offset: 1

Author

Mats Granvik, Jul 26 2010

Keywords

• nonn
• tabl

Comments

Recurrence is half of the recurrence for divisibility in A051731. That is, without subtracting (Sum_{i=1..k-1} T(n-i,k)).

Rows tend to factorial numbers.

Row sums are A177510.

Examples

			Triangle begins:
01: 1;
02: 1, 1;
03: 1, 1, 1;
04: 1, 1, 2, 1;
05: 1, 1, 2, 3,  1;
06: 1, 1, 2, 5,  4,   1;
07: 1, 1, 2, 6,  9,   5,   1;
08: 1, 1, 2, 6, 15,  14,   6,    1;
09: 1, 1, 2, 6, 20,  29,  20,    7,    1;
10: 1, 1, 2, 6, 23,  49,  49,   27,    8,    1;
11: 1, 1, 2, 6, 24,  71,  98,   76,   35,    9,    1;
12: 1, 1, 2, 6, 24,  91, 169,  174,  111,   44,   10,    1;
13: 1, 1, 2, 6, 24, 106, 259,  343,  285,  155,   54,   11,    1;
14: 1, 1, 2, 6, 24, 115, 360,  602,  628,  440,  209,   65,   12,   1;
15: 1, 1, 2, 6, 24, 119, 461,  961, 1230, 1068,  649,  274,   77,  13,   1;
16: 1, 1, 2, 6, 24, 120, 551, 1416, 2191, 2298, 1717,  923,  351,  90,  14,  1;
17: 1, 1, 2, 6, 24, 120, 622, 1947, 3606, 4489, 4015, 2640, 1274, 441, 104, 15, 1;
...
		

Crossrefs

Cf. A175105, A051731, A179749, A179750, A000142.

Programs

Sage

@CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
for n in [1..15]: print([ T(n, k) for k in [1..n] ])
# Joerg Arndt, Mar 24 2014

Formula

T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).