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A213751 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 5
1, 5, 2, 14, 9, 3, 30, 23, 13, 4, 55, 46, 32, 17, 5, 91, 80, 62, 41, 21, 6, 140, 127, 105, 78, 50, 25, 7, 204, 189, 163, 130, 94, 59, 29, 8, 285, 268, 238, 199, 155, 110, 68, 33, 9, 385, 366, 332, 287, 235, 180, 126, 77, 37, 10, 506, 485, 447, 396, 336, 271 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A051662

Antidiagonal sums:  A006325

row 1,  (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330

row 2,  (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986

row 3,  (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6

row 4,  (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6

For a guide to related arrays, see A213500.

LINKS

Table of n, a(n) for n=1..61.

FORMULA

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

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

EXAMPLE

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

1...5....14...30....55....91

2...9....23...46....80....127

3...13...32...62....105...163

4...17...41...78....130...199

5...21...50...94....155...235

6...25...59...110...180...271

MATHEMATICA

b[n_] := 2 n - 1; c[n_] := n;

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}]  (* A213751 *)

Table[t[n, n], {n, 1, 40}] (* A051662 *)

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

Table[s[n], {n, 1, 50}] (* A006325 *)

CROSSREFS

Cf. A213500.

Sequence in context: A104634 A194008 A060422 * A185781 A265434 A189236

Adjacent sequences:  A213748 A213749 A213750 * A213752 A213753 A213754

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 20 2012

STATUS

approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)