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