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%I #53 Oct 20 2024 04:08:35
%S 1,0,1,0,1,1,0,2,1,1,0,4,3,1,1,0,8,7,4,2,1,0,16,17,10,8,2,1,0,32,41,
%T 28,32,9,2,1,0,64,99,76,128,38,10,2,1,0,128,239,208,512,161,44,11,2,1,
%U 0,256,577,568,2048,682,196,50,12,3,1
%N Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.
%F For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - _Charles L. Hohn_, Aug 22 2019
%F T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - _Michel Marcus_, Aug 23 2019
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
%e 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, ...
%e 1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, ...
%e 1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...
%e 1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, ...
%e 1, 2, 10, 44, 196, 872, 3880, 17264, 76816, 341792, 1520800, ...
%e 1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, ...
%e 1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, ...
%e 1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...
%e 1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, ...
%e 1, 3, 20, 126, 796, 5028, 31760, 200616, 1267216, 8004528, 50561600, ...
%e 1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, ...
%e 1, 3, 22, 144, 952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
%e 1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
%e ...
%o (PARI) T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
%o matrix(9,9, n, k, T(n-1,k-1)) \\ _Michel Marcus_, Aug 22 2019
%o (PARI) T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
%o matrix(9, 9, n, k, T(n-1, k-1)) \\ _Charles L. Hohn_, Aug 22 2019
%Y Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
%Y Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
%Y Cf. A191348 which uses ceiling() in place of floor().
%K nonn,tabl
%O 0,8
%A _Charles L. Hohn_, May 31 2011