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A213761 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution. 5
1, 6, 4, 18, 15, 7, 40, 36, 24, 10, 75, 70, 54, 33, 13, 126, 120, 100, 72, 42, 16, 196, 189, 165, 130, 90, 51, 19, 288, 280, 252, 210, 160, 108, 60, 22, 405, 396, 364, 315, 255, 190, 126, 69, 25, 550, 540, 504, 448, 378, 300 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A172073.

Antidiagonal sums: A002419.

Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411.

Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414.

Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2.

Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2.

For a guide to related arrays, see A212500.

LINKS

Clark Kimberling, Antidiagonals n = 1..45, 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 - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4.

EXAMPLE

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

1....6....18...40....75....126

4....15...36...70....120...189

7....24...54...100...165...252

10...33...72...130...210...315

13...42...90...160...255...378

MATHEMATICA

b[n_]:=n; 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}] (* A213761 *)

Table[t[n, n], {n, 1, 40}] (* A172073 *)

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

Table[s[n], {n, 1, 50}] (* A002419 *)

CROSSREFS

Cf. A212500.

Sequence in context: A083581 A171089 A180495 * A160248 A317858 A212891

Adjacent sequences:  A213758 A213759 A213760 * A213762 A213763 A213764

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jul 04 2012

STATUS

approved

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Last modified March 20 19:57 EDT 2019. Contains 321349 sequences. (Running on oeis4.)