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A213752 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution. 5
1, 6, 3, 19, 14, 5, 44, 37, 22, 7, 85, 76, 55, 30, 9, 146, 135, 108, 73, 38, 11, 231, 218, 185, 140, 91, 46, 13, 344, 329, 290, 235, 172, 109, 54, 15, 489, 472, 427, 362, 285, 204, 127, 62, 17, 670, 651, 600, 525, 434, 335, 236, 145, 70, 19, 891, 870, 813 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A100157

Antidiagonal sums:  A071238

row 1,  (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900

row 2,  (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941

row 3,  (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6

row 4,  (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6

For a guide to related arrays, see A213500.

LINKS

Table of n, a(n) for n=1..58.

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) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.

EXAMPLE

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

1...6....19...44....85....146

3...14...37...76....135...218

5...22...55...108...185...290

7...30...73...140...235...362

9...38...91...172...285...434

MATHEMATICA

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

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}]  (* A213752 *)

Table[t[n, n], {n, 1, 40}] (* A100157 *)

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

Table[s[n], {n, 1, 50}] (* A071238 *)

CROSSREFS

Cf. A213500.

Sequence in context: A050008 A166450 A019069 * A134410 A123153 A276805

Adjacent sequences:  A213749 A213750 A213751 * A213753 A213754 A213755

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 20 2012

STATUS

approved

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)