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 A213553 #14 Sep 08 2022 08:46:02
%S A213553 1,10,8,46,43,27,146,142,118,64,371,366,334,253,125,812,806,766,658,
%T A213553 466,216,1596,1589,1541,1406,1150,775,343,2892,2884,2828,2666,2346,
%U A213553 1846,1198,512,4917,4908,4844,4655,4271,3646,2782,1753,729,7942
%N A213553 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^3, n>=1, h>=1, and ** = convolution.
%C A213553 Principal diagonal: A213554
%C A213553 Antidiagonal sums: A101089
%C A213553 row 1, (1,2,3,...)**(1,8,27,...): A024166
%C A213553 row 2, (1,2,3,...)**(8,27,64,...): (3*k^5 + 30*k^4 + 115*k^3 + 210*k^2 + 122*k)/60
%C A213553 row 3, (1,2,3,...)**(27,64,125,...): (3*k^5 + 45*k^4 + 265*k^3 + 765*k^2 + 542*k)/120
%C A213553 For a guide to related arrays, see A213500.
%H A213553 G. C. Greubel, <a href="/A213553/b213553.txt">Antidiagonals n = 1..100, flattened</a>
%F A213553 T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) -T(n,k-6).
%F A213553 G.f. for row n: f(x)/g(x), where f(x) = n^3 + (-3*n^3 + 3*n^2 + 3*n + 1)*x + (3*n^3 - 6*n^2 + 4)*x^2 - ((n-1)^3)*x^3 and g(x) = (1 - x)^6.
%F A213553 T(n,k) = k*((3*k^4 - 5*k^2 + 2) + 15*k*(k^2 - 1)*n + 30*(k^2 - 1)*n^2 + 30*(k + 1)*n^3)/60. - _G. C. Greubel_, Jul 31 2019
%e A213553 Northwest corner (the array is read by falling antidiagonals):
%e A213553 1.....10....46.....146....371
%e A213553 8.....43....142....366....806
%e A213553 27....118...334....766....1541
%e A213553 64....253...658....1406...2666
%e A213553 125...466...1150...2346...4271
%t A213553 (* First program *)
%t A213553 b[n_]:= n; c[n_]:= n^3;
%t A213553 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213553 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213553 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213553 r[n_]:= Table[T[n, k], {k, 1, 60}] (* A213553 *)
%t A213553 d = Table[T[n, n], {n, 1, 40}] (* A213554 *)
%t A213553 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213553 s1 = Table[s[n], {n, 1, 50}] (* A101089 *)
%t A213553 (* Second program *)
%t A213553 Table[Binomial[n-k+2, 2]*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30, {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 31 2019 *)
%o A213553 (PARI) t(n,k) = binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30;
%o A213553 for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 31 2019
%o A213553 (Magma) [Binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k + n*(3*n^2 +6*n +1))/30: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 31 2019
%o A213553 (Sage) [[binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k + n*(3*n^2 +6*n +1))/30 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 31 2019
%o A213553 (GAP) Flat(List([1..12], n-> List([1..n], k-> Binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30 ))); # _G. C. Greubel_, Jul 31 2019
%Y A213553 Cf. A213500.
%K A213553 nonn,tabl,easy
%O A213553 1,2
%A A213553 _Clark Kimberling_, Jun 17 2012