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A213825 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution. 4
2, 13, 8, 42, 34, 14, 98, 87, 55, 20, 190, 176, 132, 76, 26, 327, 310, 254, 177, 97, 32, 518, 498, 430, 332, 222, 118, 38, 772, 749, 669, 550, 410, 267, 139, 44, 1098, 1072, 980, 840, 670, 488, 312, 160, 50, 1505, 1476, 1372 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Principal diagonal: A213826

Antidiagonal sums: A213827

Row 1, (2,5,8,13,...)**(1,4,7,10,13,...): (3*k^2 + k)/2

Row 2, (2,5,8,13,...)**(4,7,10,13,...): (3*k^3 + 9*k^2 - 2*k)/2

Row 3, (2,5,8,13,...)**(7,10,13,16,...): (3*k^3 + 18*k^2 - 5*k)/2

For a guide to related arrays, see A212500.

LINKS

Clark Kimberling, Antidiagonals n = 1..80, flattened

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) = x*((3*n-1) + (3*n+2)*x - (6*n-8)*x^2) and g(x) = (1-x)^4.

EXAMPLE

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

2....13....42....98....190

8....34....87....176...310

14...55....132...254...430

20...76....177...332...550

26...97....222...410...670

32...118...267...488...790

MATHEMATICA

b[n_]:=3n-1; c[n_]:=3n-2;

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}] (* A213825 *)

d=Table[t[n, n], {n, 1, 40}] (* A213826 *)

d/2 (* A024215 *)

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

s1=Table[s[n], {n, 1, 50}] (* A213827 *)

CROSSREFS

Cf. A212500

Sequence in context: A158088 A124869 A292007 * A333493 A244932 A157480

Adjacent sequences:  A213822 A213823 A213824 * A213826 A213827 A213828

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jul 04 2012

STATUS

approved

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Last modified April 17 16:59 EDT 2021. Contains 343063 sequences. (Running on oeis4.)