A213778 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

1, 4, 2, 9, 6, 2, 17, 13, 7, 3, 28, 23, 15, 9, 3, 43, 37, 27, 19, 10, 4, 62, 55, 43, 33, 21, 12, 4, 86, 78, 64, 52, 37, 25, 13, 5, 115, 106, 90, 76, 58, 43, 27, 15, 5, 150, 140, 122, 106, 85, 67, 47, 31, 16, 6, 191, 180, 160, 142, 118, 97, 73, 53, 33, 18, 6, 239

Offset

1,2

Comments

Principal diagonal: A213779.

Antidiagonal sums: A213780.

Row 1, (1,2,3,4,5,...)**(1,2,2,3,3,4,4,...): A005744.

Row 2, (1,2,3,4,5,...)**(2,2,3,3,4,4,...)

Row 3, (1,2,3,4,5,...)**(3,4,4,5,5,...)

For a guide to related arrays, see A213500.

Links

Clark Kimberling, Antidiagonals n=1..80, flattened

Formula

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(1 + [n/2] + d(n)*x - [(n+1)/2]*x^2), g(x) = (1 + x)*(1 - x)^4, d(n) = (n mod 2) and [] = floor.

Example

Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...6....13...23...37...55....78
2...7....15...27...43...64....90
3...9....19...33...52...76....106
3...10...21...37...58...85....118
4...12...25...43...67...97....134
4...13...27...47...73...106...146

Mathematica

b[n_] := n; c[n_] := 1 + Floor[n/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}]  (* A213778 *)
Table[t[n, n], {n, 1, 40}] (* A213779 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213780 *)

Crossrefs

Cf. A213500.

Sequence in context: A365378 A289506 A363268 * A095833 A163990 A082156

Adjacent sequences: A213775 A213776 A213777 * A213779 A213780 A213781

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jun 21 2012

Status

approved