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%I #37 May 18 2018 12:13:28
%S 1,1,1,1,2,2,1,3,4,4,1,4,6,7,7,1,5,8,10,11,11,1,6,10,13,15,16,16,1,7,
%T 12,16,19,21,22,22,1,8,14,19,23,26,28,29,29,1,9,16,22,27,31,34,36,37,
%U 37,1,10,18,25,31,36,40,43,45,46,46,1,11,20,28,35,41
%N Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.
%C Columns are linear recurrence sequences with signature (3,-3,1).
%C 8*T(n,k) + A166147(k-1) are squares.
%C Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
%C Antidiagonals sums yield A116731.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
%F G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
%F E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
%F T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
%F T(n,k) = T(n-1,k) + n + k - 1.
%F T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
%F T(n,0) = A152947(n+1).
%F T(n,1) = A000124(n).
%F T(n,2) = A000217(n).
%F T(n,3) = A034856(n+1).
%F T(n,4) = A052905(n).
%F T(n,5) = A051936(n+4).
%F T(n,6) = A246172(n+1).
%F T(n,7) = A302537(n).
%F T(n,8) = A056121(n+1) + 1.
%F T(n,9) = A056126(n+1) + 1.
%F T(n,10) = A051942(n+10) + 1, n > 0.
%F T(n,11) = A101859(n) + 1.
%F T(n,12) = A132754(n+1) + 1.
%F T(n,13) = A132755(n+1) + 1.
%F T(n,14) = A132756(n+1) + 1.
%F T(n,15) = A132757(n+1) + 1.
%F T(n,16) = A132758(n+1) + 1.
%F T(n,17) = A212427(n+1) + 1.
%F T(n,18) = A212428(n+1) + 1.
%F T(n,n) = A143689(n) = A300192(n,2).
%F T(n,n+1) = A104249(n).
%F T(n,n+2) = T(n+1,n) = A005448(n+1).
%F T(n,n+3) = A000326(n+1).
%F T(n,n+4) = A095794(n+1).
%F T(n,n+5) = A133694(n+1).
%F T(n+2,n) = A005449(n+1).
%F T(n+3,n) = A115067(n+2).
%F T(n+4,n) = A133694(n+2).
%F T(2*n,n) = A054556(n+1).
%F T(2*n,n+1) = A054567(n+1).
%F T(2*n,n+2) = A033951(n).
%F T(2*n,n+3) = A001107(n+1).
%F T(2*n,n+4) = A186353(4*n+1) (conjectured).
%F T(2*n,n+5) = A184103(8*n+1) (conjectured).
%F T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
%F T(n,2*n) = A140066(n+1).
%F T(n+1,2*n) = A005891(n).
%F T(n+2,2*n) = A249013(5*n+4) (conjectured).
%F T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
%F T(2*n,2*n) = A143689(2*n).
%F T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
%F T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
%F T(2*n+1,2*n) = A085473(n).
%F a(n+1,5*n+1)=A051865(n+1) + 1.
%F a(n,2*n+1) = A116668(n).
%F a(2*n+1,n) = A054569(n+1).
%F T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
%F T(n,3*n) = A140063(n+1).
%F T(n+1,3*n) = A069099(n+1).
%F T(n,4*n) = A276819(n).
%F T(4*n,n) = A154106(n-1), n > 0.
%F T(2^n,2) = A028401(n+2).
%F T(1,n)*T(n,1) = A006000(n).
%F T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
%F T(n*(n+1)+1,n) = A294259(n+1).
%F T(n,n^2+1) = A081423(n).
%F T(n,A000217(n)) = A158842(n), n > 0.
%F T(n,A152947(n+1)) = A060354(n+1).
%F floor(T(n,n/2)) = A267682(n) (conjectured).
%F floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
%F floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
%F ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
%F ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
%F floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
%F ceiling(T(n,n)/n) = A007494(n), n > 0.
%F ceiling(T(n,n^2)/n) = A171769(n), n > 0.
%F ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
%F ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.
%e The array T(n,k) begins
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 ... A000012
%e 1 2 3 4 5 6 7 8 9 10 11 12 13 ... A000027
%e 2 4 6 8 10 12 14 16 18 20 22 24 26 ... A005843
%e 4 7 10 13 16 19 22 25 28 31 34 37 40 ... A016777
%e 7 11 15 19 23 27 31 35 39 43 47 51 55 ... A004767
%e 11 16 21 26 31 36 41 46 51 56 61 66 71 ... A016861
%e 16 22 28 34 40 46 52 58 64 70 76 82 88 ... A016957
%e 22 29 36 43 50 57 64 71 78 85 92 99 106 ... A016993
%e 29 37 45 53 61 69 77 85 93 101 109 117 125 ... A004770
%e 37 46 55 64 73 82 91 100 109 118 127 136 145 ... A017173
%e 46 56 66 76 86 96 106 116 126 136 146 156 166 ... A017341
%e 56 67 78 89 100 111 122 133 144 155 166 177 188 ... A017401
%e 67 79 91 103 115 127 139 151 163 175 187 199 211 ... A017605
%e 79 92 105 118 131 144 157 170 183 196 209 222 235 ... A190991
%e ...
%e The inverse binomial transforms of the columns are
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e ...
%e T(k,n-k) = A087401(n,k) + 1 as triangle
%e 1
%e 1 1
%e 1 2 2
%e 1 3 4 4
%e 1 4 6 7 7
%e 1 5 8 10 11 11
%e 1 6 10 13 15 16 16
%e 1 7 12 16 19 21 22 22
%e 1 8 14 19 23 26 28 29 29
%e 1 9 16 22 27 31 34 36 37 37
%e 1 10 18 25 31 36 40 43 45 46 46
%e ...
%p T := (n, k) -> binomial(n, 2) + k*n + 1;
%p for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
%t Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* _Michael De Vlieger_, Apr 21 2018 *)
%o (Maxima)
%o T(n, k) := binomial(n, 2)+ k*n + 1$
%o for n:0 thru 20 do
%o print(makelist(T(n, k), k, 0, 20));
%o (PARI) T(n,k) = binomial(n, 2) + k*n + 1;
%o tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, May 17 2018
%Y Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.
%K nonn,tabl
%O 0,5
%A _Franck Maminirina Ramaharo_, Apr 20 2018
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