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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). 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 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. LINKS FORMULA T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1). EXAMPLE 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; ... PROG (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)) (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] ] # print triangle # Joerg Arndt, Mar 24 2014 CROSSREFS Cf. A175105, A051731, A179749, A179750, A000142. Sequence in context: A054124 A144406 A238888 * A096670 A130461 A225631 Adjacent sequences:  A179745 A179746 A179747 * A179749 A179750 A179751 KEYWORD nonn,tabl AUTHOR Mats Granvik, Jul 26 2010 STATUS approved

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Last modified January 26 14:08 EST 2020. Contains 331280 sequences. (Running on oeis4.)