A213548 - OEIS

%I #14 Oct 05 2016 08:39:24

%S 1,5,3,15,12,6,35,31,22,10,70,65,53,35,15,126,120,105,81,51,21,210,

%T 203,185,155,115,70,28,330,322,301,265,215,155,92,36,495,486,462,420,

%U 360,285,201,117,45,715,705,678,630,560,470,365,253,145,55,1001

%N 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 Principal diagonal: A213549.

%C Antidiagonal sums: A051836.

%C Row 1, (1,2,3,...)**(1,3,6,...): A000332.

%C Row 2, (1,2,3,...)**(3,6,10,...): A005718.

%C Row 3, (1,2,3,...)**(6,10,15,...): k*(k+1)*(k^2 + 13*k + 58)/24.

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

%H Clark Kimberling, <a href="/A213548/b213548.txt">Antidiagonals n = 1..60</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*(n+1) - 2*((n-1)^2)*x + n*(n-1)*x^2 and g(x) = 2*(1 - x)^5.

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

%e .  1,  5,  15,  35,  70, ...

%e .  3, 12,  31,  65, 120, ...

%e .  6, 22,  53, 105, 185, ...

%e . 10, 35,  81, 155, 265, ...

%e . 15, 51, 115, 215, 360, ...

%e . 21, 70, 155, 285, 470, ...

%e ...

%e T(5,1) = (1)**(15) = 15;

%e T(5,2) = (1,2)**(15,21) = 1*21 + 2*15 = 51;

%e T(5,3) = (1,2,3)**(15,21,28) = 1*28 + 2*21 + 3*15 = 115;

%e T(4,4) = (1,2,3,4)**(10,15,21,28) = 1*28 + 2*21 + 3*15 + 4*10 = 155.

%t b[n_] := n; c[n_] := n (n + 1)/2

%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}]]

%t r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213548 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A213549 *)

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

%t s1 = Table[s[n], {n, 1, 50}] (* A051836 *)

%Y Cf. A213500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 16 2012