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%I #61 Apr 10 2019 21:54:35
%S 0,1,1,2,2,2,3,4,4,3,4,7,8,7,4,5,11,14,14,11,5,6,16,22,24,22,16,6,7,
%T 22,32,37,37,32,22,7,8,29,44,53,56,53,44,29,8,9,37,58,72,79,79,72,58,
%U 37,9,10,46,74,94,106,110,106,94,74,46,10,11,56,92,119
%N Array T(n,k) = n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1) read by antidiagonals.
%C Antidiagonal sums are given by 2*A055795.
%C Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
%C Some primes in the array are
%C n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
%C n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
%C n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.
%D Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
%D L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
%D R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.
%H Michael De Vlieger, <a href="/A300401/b300401.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H Cheyne Homberger, <a href="https://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%F T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
%F T(n,0) = A001477(n).
%F T(n,1) = A000124(n).
%F T(n,2) = A014206(n).
%F T(n,3) = A273465(3*n+2).
%F T(n,4) = A084849(n+1).
%F T(n,n) = A179000(n-1,n), n >= 1.
%F T(2*n,2*n) = 8*A081436(n-1), n >= 1.
%F T(2*n+1,2*n+1) = 2*A006000(2*n+1).
%F T(n,n+1) = A188377(n+3).
%F T(n,n+2) = A188377(n+2), n >= 1.
%F Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
%F G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
%F E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).
%e The array T(n,k) begins
%e 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e 1 2 4 7 11 16 22 29 37 46 56 67 ...
%e 2 4 8 14 22 32 44 58 74 92 112 134 ...
%e 3 7 14 24 37 53 72 94 119 147 178 212 ...
%e 4 11 22 37 56 79 106 137 172 211 254 301 ...
%e 5 16 32 53 79 110 146 187 233 284 340 401 ...
%e 6 22 44 72 106 146 192 244 302 366 436 512 ...
%e 7 29 58 94 137 187 244 308 379 457 542 634 ...
%e 8 37 74 119 172 233 302 379 464 557 658 767 ...
%e 9 46 92 147 211 284 366 457 557 666 784 911 ...
%e 10 56 112 178 254 340 436 542 658 784 920 1066 ...
%e 11 67 134 212 301 401 512 634 767 911 1066 1232 ...
%e 12 79 158 249 352 467 594 733 884 1047 1222 1409 ...
%e 13 92 184 289 407 538 682 839 1009 1192 1388 1597 ...
%e 14 106 212 332 466 614 776 952 1142 1346 1564 1796 ...
%e 15 121 242 378 529 695 876 1072 1283 1509 1750 2006 ...
%e 16 137 274 427 596 781 982 1199 1432 1681 1946 2227 ...
%e 17 154 308 479 667 872 1094 1333 1589 1862 2152 2459 ...
%e 18 172 344 534 742 968 1212 1474 1754 2052 2368 2702 ...
%e 19 191 382 592 821 1069 1336 1622 1927 2251 2594 2956 ...
%e 20 211 422 653 904 1175 1466 1777 2108 2459 2830 3221 ...
%e ...
%e The inverse binomial transforms of the columns are
%e 0 1 2 3 4 5 6 7 8 9 10 11 ... A001477
%e 1 1 2 4 7 11 22 29 37 45 56 67 ... A152947
%e 0 1 2 3 4 5 6 7 8 9 10 11 ... A001477
%e 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 ...
%e 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e ...
%p T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
%p for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
%t T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
%t Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 07 2018 *)
%o (Maxima)
%o T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
%o for n:0 thru 20 do
%o print(makelist(T(n, k), k, 0, 20));
%o (PARI) T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
%o tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Mar 12 2018
%Y Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.
%K nonn,tabl
%O 0,4
%A _Franck Maminirina Ramaharo_, Mar 05 2018