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A213828
Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.
4
2, 13, 5, 42, 28, 8, 98, 78, 43, 11, 190, 164, 114, 58, 14, 327, 295, 230, 150, 73, 17, 518, 480, 400, 296, 186, 88, 20, 772, 728, 633, 505, 362, 222, 103, 23, 1098, 1048, 938, 786, 610, 428, 258, 118, 26, 1505, 1449, 1324
OFFSET
1,1
COMMENTS
Principal diagonal: A213829.
Antidiagonal sums: A213830.
Row 1, (1,4,7,10,...)**(2,5,8,11,...): (3*k^2 + k)/2.
Row 2, (1,4,7,10,...)**(5,8,11,14,...): (3*k^3 + 9*k^2 + 4*k)/2.
Row 3, (1,4,7,10,...)**(8,11,14,17,...): (3*k^3 + 18*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.
LINKS
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*((6*n-4) - (3*n-8)*x - (3*n-5)*x^2) and g(x) = (1-x)^4.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
2....13...42....98....190
5....28...78....164...295
8....43...114...230...400
11...58...150...296...505
14...73...186...362...610
17...88...222...428...715
MATHEMATICA
b[n_]:=3n-2; c[n_]:=3n-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}] (* A213828 *)
d=Table[t[n, n], {n, 1, 40}] (* A213829 *)
d/2 (* A005915 *)
s[n_]:=Sum[t[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213830 *)
CROSSREFS
Sequence in context: A084160 A238139 A268722 * A324003 A176466 A128155
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Jul 04 2012
STATUS
approved