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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).

%I A179748 #27 Mar 07 2020 11:42:32
%S A179748 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,
%T A179748 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,
%U A179748 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
%N 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).
%C A179748 Recurrence is half of the recurrence for divisibility in A051731. That is, without subtracting (Sum_{i=1..k-1} T(n-i,k)).
%C A179748 Rows tend to factorial numbers.
%C A179748 Row sums are A177510.
%F A179748 T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).
%e A179748 Triangle begins:
%e A179748 01: 1;
%e A179748 02: 1, 1;
%e A179748 03: 1, 1, 1;
%e A179748 04: 1, 1, 2, 1;
%e A179748 05: 1, 1, 2, 3,  1;
%e A179748 06: 1, 1, 2, 5,  4,   1;
%e A179748 07: 1, 1, 2, 6,  9,   5,   1;
%e A179748 08: 1, 1, 2, 6, 15,  14,   6,    1;
%e A179748 09: 1, 1, 2, 6, 20,  29,  20,    7,    1;
%e A179748 10: 1, 1, 2, 6, 23,  49,  49,   27,    8,    1;
%e A179748 11: 1, 1, 2, 6, 24,  71,  98,   76,   35,    9,    1;
%e A179748 12: 1, 1, 2, 6, 24,  91, 169,  174,  111,   44,   10,    1;
%e A179748 13: 1, 1, 2, 6, 24, 106, 259,  343,  285,  155,   54,   11,    1;
%e A179748 14: 1, 1, 2, 6, 24, 115, 360,  602,  628,  440,  209,   65,   12,   1;
%e A179748 15: 1, 1, 2, 6, 24, 119, 461,  961, 1230, 1068,  649,  274,   77,  13,   1;
%e A179748 16: 1, 1, 2, 6, 24, 120, 551, 1416, 2191, 2298, 1717,  923,  351,  90,  14,  1;
%e A179748 17: 1, 1, 2, 6, 24, 120, 622, 1947, 3606, 4489, 4015, 2640, 1274, 441, 104, 15, 1;
%e A179748 ...
%o A179748 (Excel cell formula European dot comma style) =if(column()=1; 1; if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4)); 0))
%o A179748 (Sage)
%o A179748 @CachedFunction
%o A179748 def T(n, k): # A179748
%o A179748     if n == 0:  return int(k==0);
%o A179748     if k == 1:  return int(n>=1);
%o A179748     return sum( T(n-i, k-1) for i in [1..k-1] );
%o A179748 for n in [1..15]: print([ T(n, k) for k in [1..n] ])
%o A179748 # _Joerg Arndt_, Mar 24 2014
%Y A179748 Cf. A175105, A051731, A179749, A179750, A000142.
%K A179748 nonn,tabl
%O A179748 1,9
%A A179748 _Mats Granvik_, Jul 26 2010