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%I #22 Sep 08 2022 08:45:01
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,2,3,3,1,1,1,1,2,3,4,
%T 3,1,1,1,1,2,3,5,6,4,1,1,1,1,2,3,5,7,7,4,1,1,1,1,2,3,5,8,11,10,5,1,1,
%U 1,1,2,3,5,8,12,14,11,5,1,1,1,1,2,3,5,8,13,19,21,15,6,1
%N Triangle T read by rows: T(i,0)=T(i,i)=1, T(i,j) = Sum_{k=1..floor(n/2)} T(i-2k, j-2k+1) for 1<=j<=i-1, where T(m,n) := 0 if m<0 or n<0.
%C T(i+j,j) is the number of strings (s(1),...,s(m)) of nonnegative integers s(k) such that m<=i+1, s(m)=j and s(k)-s(k-1) is an odd positive integer for k=2,3,...,m.
%C T(i+j,j) is the number of compositions of numbers <=j using up to i parts, each an odd positive integer.
%H G. C. Greubel, <a href="/A055801/b055801.txt">Rows n = 0..100 of triangle, flattened</a>
%H Clark Kimberling, <a href="https://www.fq.math.ca/Scanned/40-4/kimberling.pdf">Path-counting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328-338, Example 2B.
%e Rows:
%e 1
%e 1 1
%e 1 1 1
%e 1 1 1 1
%e 1 1 1 2 1
%e 1 1 1 2 2 1
%e 1 1 1 2 3 3 1
%e 1 1 1 2 3 4 3 1
%e 1 1 1 2 3 5 6 4 1
%e 1 1 1 2 3 5 7 7 4 1
%e 1 1 1 2 3 5 8 11 10 5 1
%e 1 1 1 2 3 5 8 12 14 11 5 1
%e 1 1 1 2 3 5 8 13 19 21 15 6 1
%e 1 1 1 2 3 5 8 13 20 26 25 16 6 1
%e 1 1 1 2 3 5 8 13 21 32 40 36 21 7 1
%e 1 1 1 2 3 5 8 13 21 33 46 51 41 22 7 1
%e T(9,6) counts the strings 3456, 1236, 1256, 1456, 036, 016, 056.
%e T(9,6) counts the compositions 111, 113, 131, 311, 33, 15, 51.
%p A055801 := proc(i,j) option remember;
%p if j =0 or j = i then 1;
%p elif i < 0 or j < 0 then 0;
%p else add(procname(i-2*k,j-2*k+1),k=1..floor(i/2)) ;
%p end if;
%p end proc:
%p seq(seq(A055801(n,k), k=0..n),n=0..20); # _R. J. Mathar_, Feb 11 2018
%t T[n_, k_]:= T[n, k]= If[n<0 || k<0, 0, If[k==0 || k==n, 1, Sum[T[n-2*j, k-2*j+1 ], {j, Floor[n/2]}]]]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 23 2020 *)
%o (PARI) T(n,k) = if(n<0 || k<0, 0, if(k==0 || k==n, 1, sum(j=1, n\2, T(n-2*j, k-2*j+1))));
%o for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jan 23 2020
%o (Magma)
%o function T(n,k)
%o if n lt 0 or k lt 0 then return 0;
%o elif k eq 0 or k eq n then return 1;
%o else return (&+[T(n-2*j, k-2*j+1): j in [1..Floor(n/2)]]);
%o end if; return T; end function;
%o [T(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jan 23 2020
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (n<0 or k<0): return 0
%o elif (k==0 or k==n): return 1
%o else: return sum(T(n-2*j, k-2*j+1) for j in (1..floor(n/2)))
%o [[T(n, k) for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, Jan 23 2020
%o (GAP)
%o T:= function(n,k)
%o if n<0 or k<0 then return 0;
%o elif k=0 or k=n then return 1;
%o else return Sum([1..Int(n/2)], j-> T(n-2*j, k-2*j+1));
%o fi; end;
%o Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Jan 23 2020
%Y Infinitely many of the columns are (1, 1, 1, 2, 3, 5, 8, ..., Fibonacci numbers)
%Y Essentially a reflected version of A011794.
%Y Cf. A055802, A055803, A055804, A055805, A055806.
%K nonn,tabl,easy
%O 0,14
%A _Clark Kimberling_, May 28 2000
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