This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213582 #15 May 09 2025 23:08:26
%S A213582 1,5,2,16,9,3,42,27,13,4,99,68,38,17,5,219,156,94,49,21,6,466,339,213,
%T A213582 120,60,25,7,968,713,459,270,146,71,29,8,1981,1470,960,579,327,172,82,
%U A213582 33,9,4017,2994,1972,1207,699,384,198,93,37,10,8100,6053,4007,2474,1454,819,441,224,104,41,11
%N A213582 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213582 Principal diagonal: A213583.
%C A213582 Antidiagonal sums: A156928.
%C A213582 Row 1, (1,3,7,15,31,...)**(1,2,3,4,5,...): A002662.
%C A213582 Row 2, (1,3,7,15,31,...)**(2,3,4,5,6,...)
%C A213582 Row 3, (1,3,7,15,31,...)**(3,4,5,6,7,...)
%C A213582 For a guide to related arrays, see A213500.
%H A213582 Clark Kimberling, <a href="/A213582/b213582.txt">Antidiagonals n = 1..60, flattened</a>
%F A213582 T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
%F A213582 G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-2*x) *(1-x)^3.
%F A213582 T(n,k) = 2*(n+1)*(2^k - 1) - k*(k + 2*n + 3)/2. - _G. C. Greubel_, Jul 08 2019
%e A213582 Northwest corner (the array is read by falling antidiagonals):
%e A213582 1...5....16...42....99....219
%e A213582 2...9....27...68....156...339
%e A213582 3...13...38...94....213...459
%e A213582 4...17...49...120...270...579
%e A213582 5...21...60...146...327...699
%e A213582 6...25...71...172...384...819
%t A213582 (* First program *)
%t A213582 b[n_]:= 2^n - 1; c[n_]:= n;
%t A213582 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213582 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213582 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
%t A213582 r[n_]:= Table[T[n, k], {k, 40}]
%t A213582 Table[T[n, n], {n, 1, 40}] (* A213583 *)
%t A213582 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213582 Table[s[n], {n, 1, 50}] (* A156928 *)
%t A213582 (* Second program *)
%t A213582 Table[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2, {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 08 2019 *)
%o A213582 (PARI) t(n,k) = 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2;
%o A213582 for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 08 2019
%o A213582 (Magma) [[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2: k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jul 08 2019
%o A213582 (Sage) [[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 08 2019
%o A213582 (GAP) Flat(List([1..12], n-> List([1..n], k-> 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 ))); # _G. C. Greubel_, Jul 08 2019
%Y A213582 Cf. A213500, A213571.
%K A213582 nonn,tabl,easy
%O A213582 1,2
%A A213582 _Clark Kimberling_, Jun 19 2012