A213773 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

1, 8, 4, 30, 23, 7, 76, 66, 38, 10, 155, 142, 102, 53, 13, 276, 260, 208, 138, 68, 16, 448, 429, 365, 274, 174, 83, 19, 680, 658, 582, 470, 340, 210, 98, 22, 981, 956, 868, 735, 575, 406, 246, 113, 25, 1360, 1332, 1232, 1078, 888, 680, 472, 282, 128, 28

Offset

1,2

Comments

Principal diagonal: A213782

Antidiagonal sums: A214092

Row 1, (1,4,7,10,...)**(1,4,7,10,...): A100175

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

Row 3, (1,4,7,10,...)**(7,10,13,16,...): (3*k^3 + 15*k^2 - 4*k)/2

For a guide to related arrays, see A213500.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1034

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+1)*x - (6*n-10)*x^2) and g(x) = (1-x)^4.

Example

Northwest corner (the array is read by falling antidiagonals):
  1    8    30    76    155   276
  4    23   66    142   260   429
  7    38   102   208   365   582
  10   53   138   274   470   735
  13   68   174   340   575   888

Mathematica

b[n_]:=3n-2; 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}] (* A213773 *)
Table[t[n, n], {n, 1, 40}] (* A214092 *)
s[n_]:=Sum[t[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213818 *)

Crossrefs

Cf. A100175, A213500, A213773, A213782, A213818, A214092.

Sequence in context: A160411 A033473 A238163 * A213178 A082682 A279635

Adjacent sequences: A213770 A213771 A213772 * A213774 A213775 A213776

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jul 04 2012

Status

approved