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A213503 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213503 #15 Jan 09 2025 18:39:19
%S A213503 1,6,2,20,11,3,50,34,16,4,105,80,48,21,5,196,160,110,62,26,6,336,287,
%T A213503 215,140,76,31,7,540,476,378,270,170,90,36,8,825,744,616,469,325,200,
%U A213503 104,41,9,1210,1110,948,756,560,380,230,118,46,10
%N A213503 Rectangular array:  (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213503 Principal diagonal:  A117066
%C A213503 Antidiagonal sums:  A033455
%C A213503 For a guide to related arrays, see A213500.
%H A213503 G. C. Greubel, <a href="/A213503/b213503.txt">Antidiagonal rows n = 1..100</a>
%F A213503 T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).
%F A213503 G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)^2 x^2 and g(x) = (1 - x)^5.
%F A213503 T(n,k) = k*(k^3 + 4*k^2*n + 6*k*n - k + 2*n)/12. - _G. C. Greubel_, Jul 05 2019
%e A213503 Northwest corner (the array is read by falling antidiagonals):
%e A213503 1....6....20....50....105....196...336
%e A213503 2....11...34....80....160....287...476
%e A213503 3....16...48....110...215....378...616
%e A213503 4....21...62....140...270....469...756
%e A213503 5....26...76....170...325....560...896
%e A213503 ...
%e A213503 T(5,1) = (1)**(5) = 5
%e A213503 T(5,2) = (1,4)**(5,6) = 1*6+4*5 = 26
%e A213503 T(5,3) = (1,4,9)**(5,6,7) = 1*7+4*6+9*5 = 76
%t A213503 (* First program *)
%t A213503 b[n_]:= n^2; c[n_]:= n;
%t A213503 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213503 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213503 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213503 *)
%t A213503 r[n_]:= Table[T[n, k], {k,40}]  (* columns of antidiagonal triangle *)
%t A213503 d = Table[T[n, n], {n, 1, 40}] (* A117066 *)
%t A213503 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213503 s1 = Table[s[n], {n, 1, 50}] (* A033455 *)
%t A213503 (* Second program *)
%t A213503 Table[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12, {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 05 2019 *)
%o A213503 (PARI) t(n,k) = (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12;
%o A213503 for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 05 2019
%o A213503 (Magma) [[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12: k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jul 05 2019
%o A213503 (Sage) [[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 05 2019
%o A213503 (GAP) Flat(List([1..12], n-> List([1..n], k-> (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12))); # _G. C. Greubel_, Jul 05 2019
%Y A213503 Cf. A213500.
%K A213503 nonn,easy,tabl
%O A213503 1,2
%A A213503 _Clark Kimberling_, Jun 16 2012