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A213590 Rectangular array:  (row n) = b**c, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution. 6
1, 5, 1, 15, 6, 2, 36, 20, 11, 3, 76, 51, 35, 17, 5, 148, 112, 87, 55, 28, 8, 273, 224, 188, 138, 90, 45, 13, 485, 421, 372, 300, 225, 145, 73, 21, 839, 758, 694, 596, 488, 363, 235, 118, 34, 1424, 1324, 1243, 1115, 968, 788, 588, 380, 191, 55, 2384 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213504.

Antidiagonal sums: A213557.

Row 1,  (1,4,9,16,...)**(1,1,2,3,5,...): A053808.

Row 2,  (1,4,9,16,...)**(1,2,3,5,8,...).

Row 3,  (1,4,9,16,...)**(2,3,5,8,13,...).

For a guide to related arrays, see A213500.

LINKS

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

FORMULA

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

Columns:  T(n,k)=T(n-1,k) + T(n-2,k).

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

EXAMPLE

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

1....5....15....36....76.....148

1....6....20....51....112....224

2....11...35....87....188....372

3....17...55....138...300....596

5....28...90....225...488....868

8....45...145...363...788....1564

13...73...235...588...1276...2532

MATHEMATICA

b[n_] := n^2; 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}]  (* A213590 *)

Table[t[n, n], {n, 1, 40}] (* A213504 *)

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

Table[s[n], {n, 1, 50}] (* A213557 *)

CROSSREFS

Cf. A213500, A213587.

Sequence in context: A295574 A087727 A039807 * A185263 A264616 A157395

Adjacent sequences:  A213587 A213588 A213589 * A213591 A213592 A213593

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 19 2012

STATUS

approved

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Last modified February 21 16:41 EST 2018. Contains 299414 sequences. (Running on oeis4.)