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A213568 Rectangular array:  (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 8
1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A213569

Antidiagonal sums:  A047520

Row 1,  (1,3,6,...)**(1,4,9,...):  A125128

Row 2,  (1,3,6,...)**(4,9,16,...):  A095151

Row 3,  (1,3,6,...)**(9,16,25,...):  A000247

Row 4,  (1,3,6,...)**(16,25,36...):  A208638 (?)

For a guide to related arrays, see A213500.

LINKS

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

FORMULA

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

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

EXAMPLE

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

1...4....11...26....57....120

2...7....18...41....88....183

3...10...25...56....119...246

4...13...32...71....150...309

5...16...39...86....181...372

6...19...46...101...212...435

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}]  (* A213568 *)

d = Table[t[n, n], {n, 1, 40}] (* A213569 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A047520 *)

CROSSREFS

Cf. A213500.

Sequence in context: A191434 A163918 A154699 * A185878 A182870 A094406

Adjacent sequences:  A213565 A213566 A213567 * A213569 A213570 A213571

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 18 2012

STATUS

approved

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Last modified October 18 10:28 EDT 2017. Contains 293507 sequences.