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%I #13 Apr 07 2020 23:00:24
%S 2,3,4,5,7,6,9,13,11,8,17,25,21,15,10,33,49,41,29,19,12,65,97,81,57,
%T 37,23,14,129,193,161,113,73,45,27,16,257,385,321,225,145,89,53,31,18,
%U 513,769,641,449,289,177,105,61,35,20,1025,1537,1281,897,577,353,209,121,69,39,22
%N Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.
%C This entry was motivated by a class work of Ferran D.
%F Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
%F Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
%F O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
%F E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
%F E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.
%e The array A begins:
%e k\n 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 2 3 5 9 17 33 65 129 257 513 1025
%e 1: 4 7 13 25 49 97 193 385 769 1537 3073
%e 2: 6 11 21 41 81 161 321 641 1281 2561 5121
%e 3: 8 15 29 57 113 225 449 897 1793 3585 7169
%e 4: 10 19 37 73 145 289 577 1153 2305 4609 9217
%e 5: 12 23 45 89 177 353 705 1409 2817 5633 11265
%e 6: 14 27 53 105 209 417 833 1665 3329 6657 13313
%e 7: 16 31 61 121 241 481 961 1921 3841 7681 15361
%e 8: 18 35 69 137 273 545 1089 2177 4353 8705 17409
%e 9: 20 39 77 153 305 609 1217 2433 4865 9729 19457
%e ...
%e The triangle T begins:
%e m\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 2
%e 1: 3 4
%e 2: 5 7 6
%e 3: 9 13 11 8
%e 4: 17 25 21 15 10
%e 5: 33 49 41 29 19 12
%e 6: 65 97 81 57 37 23 14
%e 7: 129 193 161 113 73 45 27 16
%e 8: 257 385 321 225 145 89 53 31 18
%e 9: 513 769 641 449 289 177 105 61 35 20
%e 10: 1025 1537 1281 897 577 353 209 121 69 39 22
%e ...
%t Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* _Michael De Vlieger_, Jun 25 2017 *)
%o (PARI) A(n, k) = (2*n + 1)*2^k + 1;
%o for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ _Indranil Ghosh_, Jun 22 2017
%Y Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.
%K nonn,tabl,easy
%O 0,1
%A _Wolfdieter Lang_, Jun 21 2017
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