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A213765 Rectangular array:  (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution. 4
1, 4, 1, 10, 5, 2, 21, 14, 9, 3, 40, 31, 24, 14, 5, 72, 61, 52, 38, 23, 8, 125, 112, 101, 83, 62, 37, 13, 212, 197, 184, 162, 135, 100, 60, 21, 354, 337, 322, 296, 263, 218, 162, 97, 34, 585, 566, 549, 519, 480, 425, 353, 262, 157, 55, 960, 939, 920, 886 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213766.

Antidiagonal sums: A213767.

Row 1,  (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.

Row 2,  (1,3,5,7,9,...)**(1,2,3,5,8,...): A023652.

Row 3,  (1,3,5,7,9,...)**(2,3,5,8,13,...).

For a guide to related arrays, see A213500.

LINKS

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

FORMULA

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

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

T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - Ehren Metcalfe, Jul 10 2019

EXAMPLE

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

1....4....10....21....40....72

1....5....14....31....61....112

2....9....24....52....101...184

3....14...38....83....162...296

5....23...62....135...263...480

8....37...100...218...425...776

13...60...162...353...688...1256

MATHEMATICA

b[n_] := 2 n - 1; c[n_] := Fibonacci[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}]  (* A213765 *)

Table[t[n, n], {n, 1, 40}] (* A213766 *)

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

Table[s[n], {n, 1, 50}] (* A213767 *)

CROSSREFS

Cf. A213500.

Sequence in context: A186368 A185676 A277583 * A182971 A062145 A178216

Adjacent sequences:  A213762 A213763 A213764 * A213766 A213767 A213768

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 21 2012

STATUS

approved

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Last modified February 28 01:15 EST 2020. Contains 332319 sequences. (Running on oeis4.)