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Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
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%I #15 Jan 09 2025 18:39:19

%S 1,6,2,20,11,3,50,34,16,4,105,80,48,21,5,196,160,110,62,26,6,336,287,

%T 215,140,76,31,7,540,476,378,270,170,90,36,8,825,744,616,469,325,200,

%U 104,41,9,1210,1110,948,756,560,380,230,118,46,10

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

%C Principal diagonal: A117066

%C Antidiagonal sums: A033455

%C For a guide to related arrays, see A213500.

%H G. C. Greubel, <a href="/A213503/b213503.txt">Antidiagonal rows n = 1..100</a>

%F 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 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 T(n,k) = k*(k^3 + 4*k^2*n + 6*k*n - k + 2*n)/12. - _G. C. Greubel_, Jul 05 2019

%e Northwest corner (the array is read by falling antidiagonals):

%e 1....6....20....50....105....196...336

%e 2....11...34....80....160....287...476

%e 3....16...48....110...215....378...616

%e 4....21...62....140...270....469...756

%e 5....26...76....170...325....560...896

%e ...

%e T(5,1) = (1)**(5) = 5

%e T(5,2) = (1,4)**(5,6) = 1*6+4*5 = 26

%e T(5,3) = (1,4,9)**(5,6,7) = 1*7+4*6+9*5 = 76

%t (* First program *)

%t b[n_]:= n^2; c[n_]:= n;

%t T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]

%t TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]

%t Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213503 *)

%t r[n_]:= Table[T[n, k], {k,40}] (* columns of antidiagonal triangle *)

%t d = Table[T[n, n], {n, 1, 40}] (* A117066 *)

%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]

%t s1 = Table[s[n], {n, 1, 50}] (* A033455 *)

%t (* Second program *)

%t 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 (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 for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 05 2019

%o (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 (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 (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 Cf. A213500.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Jun 16 2012