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%I #13 May 07 2019 17:38:24
%S 1,1,1,1,4,1,1,16,16,1,1,67,79,67,1,1,281,344,344,281,1,1,1176,1453,
%T 1504,1453,1176,1,1,4921,6093,6358,6358,6093,4921,1,1,20594,25511,
%U 26671,26885,26671,25511,20594,1,1,86185,106775,111680,112789,112789,111680,106775,86185,1
%N Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 1, where b(n) = 5*b(n-1) - 4*b(n-2) + 3*b(n-3) - 2*b(n-4) - b(n-5) and b(0) = 0, b(1) = 1, b(2) = 5, b(3) = 21, b(4) = 88.
%C Row sums are {1, 2, 6, 34, 215, 1252, 6764, 34746, 172439, 834860, 3967727, ...}.
%H Indranil Ghosh, <a href="/A176483/b176483.txt">Rows 0..120, flattened</a>
%H Indranil Ghosh, <a href="/A176483/a176483.txt">Python Program to generate the b-file</a>
%F Let b(n) = 5*b(n-1) - 4*b(n-2) + 3*b(n-3) - 2*b(n-4) - b(n-5), with b(0) = 0, b(1) = 1, b(2) = 5, b(3) = 21, b(4) = 88, then T(n, k) = b(n) - b(k) - b(n-k) + 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 16, 16, 1;
%e 1, 67, 79, 67, 1;
%e 1, 281, 344, 344, 281, 1;
%e 1, 1176, 1453, 1504, 1453, 1176, 1;
%e 1, 4921, 6093, 6358, 6358, 6093, 4921, 1;
%e 1, 20594, 25511, 26671, 26885, 26671, 25511, 20594, 1;
%e 1, 86185, 106775, 111680, 112789, 112789, 111680, 106775, 86185, 1;
%e ...
%e T(3,2) = b(3) - b(2) - b(3 - 2) + 1 = 21 - 5 - 1 + 1 = 16 [b(1) = 1, b(2) = 5, b(3) = 21]. - _Indranil Ghosh_, Feb 17 2017
%t b[0]:=0; b[1]:=1; b[2]:=5; b[3]:=21; b[4]:=88;
%t b[n_]:= 5*b[n-1] -4*b[n-2] +3*b[n-3] -2*b[n-4] -b[n-5];
%t T[n_, m_]:= b[n] -b[m] -b[n-m] +1;
%t Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten (* modified by _G. C. Greubel_, May 06 2019 *)
%o (PARI)
%o {b(n) = if(n==0, 0, if(n==1, 1, if(n==2, 5, if(n==3, 21, if(n==4, 88, 5*b(n-1) -4*b(n-2) +3*b(n-3) -2*b(n-4) -b(n-5))))))};
%o {T(n, k) = b(n) -b(k) -b(n-k) +1};
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 06 2019
%o (Sage)
%o def b(n):
%o if (n==0): return 0
%o elif (n==1): return 1
%o elif (n==2): return 5
%o elif (n==3): return 21
%o elif (n==4): return 88
%o else: return 5*b(n-1) -4*b(n-2) +3*b(n-3) -2*b(n-4) -b(n-5)
%o def T(n, k): return b(n) - b(k) - b(n-k) + 1
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 06 2019
%Y Cf. A095263.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Apr 18 2010
%E Edited by _G. C. Greubel_, May 06 2019
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