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A213548 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = m(m+1)/2, m = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213548 #14 Oct 05 2016 08:39:24
%S A213548 1,5,3,15,12,6,35,31,22,10,70,65,53,35,15,126,120,105,81,51,21,210,
%T A213548 203,185,155,115,70,28,330,322,301,265,215,155,92,36,495,486,462,420,
%U A213548 360,285,201,117,45,715,705,678,630,560,470,365,253,145,55,1001
%N A213548 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = m(m+1)/2, m = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213548 Principal diagonal: A213549.
%C A213548 Antidiagonal sums: A051836.
%C A213548 Row 1, (1,2,3,...)**(1,3,6,...): A000332.
%C A213548 Row 2, (1,2,3,...)**(3,6,10,...): A005718.
%C A213548 Row 3, (1,2,3,...)**(6,10,15,...): k*(k+1)*(k^2 + 13*k + 58)/24.
%C A213548 For a guide to related arrays, see A213500.
%H A213548 Clark Kimberling, <a href="/A213548/b213548.txt">Antidiagonals n = 1..60</a>
%F A213548 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 A213548 G.f. for row n:  f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + n*(n-1)*x^2 and g(x) = 2*(1 - x)^5.
%e A213548 Northwest corner (the array is read by falling antidiagonals):
%e A213548 .  1,  5,  15,  35,  70, ...
%e A213548 .  3, 12,  31,  65, 120, ...
%e A213548 .  6, 22,  53, 105, 185, ...
%e A213548 . 10, 35,  81, 155, 265, ...
%e A213548 . 15, 51, 115, 215, 360, ...
%e A213548 . 21, 70, 155, 285, 470, ...
%e A213548 ...
%e A213548 T(5,1) = (1)**(15) = 15;
%e A213548 T(5,2) = (1,2)**(15,21) = 1*21 + 2*15 = 51;
%e A213548 T(5,3) = (1,2,3)**(15,21,28) = 1*28 + 2*21 + 3*15 = 115;
%e A213548 T(4,4) = (1,2,3,4)**(10,15,21,28) = 1*28 + 2*21 + 3*15 + 4*10 = 155.
%t A213548 b[n_] := n; c[n_] := n (n + 1)/2
%t A213548 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213548 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213548 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213548 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213548 *)
%t A213548 d = Table[t[n, n], {n, 1, 40}] (* A213549 *)
%t A213548 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213548 s1 = Table[s[n], {n, 1, 50}] (* A051836 *)
%Y A213548 Cf. A213500.
%K A213548 nonn,tabl,easy
%O A213548 1,2
%A A213548 _Clark Kimberling_, Jun 16 2012