A213561 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = m*(m+1)/2, m=n-1+h, n>=1, h>=1, and ** = convolution.

1, 7, 3, 27, 18, 6, 77, 61, 34, 10, 182, 157, 109, 55, 15, 378, 342, 267, 171, 81, 21, 714, 665, 557, 407, 247, 112, 28, 1254, 1190, 1043, 827, 577, 337, 148, 36, 2079, 1998, 1806, 1512, 1152, 777, 441, 189, 45, 3289, 3189, 2946, 2562, 2072, 1532

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213562

Antidiagonal sums: A213563

Row 1, (1,4,9,...)**(1,3,6,...): A005585

Row 2, (1,4,9,...)**(3,6,10,...): (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120

Row 3, (1,4,9,...)**(6,10,15,...): (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
3....18....61....157...342
6....34....109...267...557
10...55....171...407...827
15...81....247...577...1152
21...112...337...777...1532
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := n^2; c[n_] := n (n + 1)/2
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213561 *)
d = Table[t[n, n], {n, 1, 40}] (* A213562 *)
s1 = Table[s[n], {n, 1, 50}] (* A213563 *)

Formula

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).

G.f. for row n: f(x)/g(x), where f(x) = n*(n + 1) - (n^2 - n - 2)*x - (n^2 + n - 2)*x^2 + n*(n - 1)*x^3 and g(x) = 2*(1 - x)^6.