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A213564
Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
5
1, 7, 4, 27, 21, 9, 77, 67, 43, 16, 182, 167, 127, 73, 25, 378, 357, 297, 207, 111, 36, 714, 686, 602, 467, 307, 157, 49, 1254, 1218, 1106, 917, 677, 427, 211, 64, 2079, 2034, 1890, 1638, 1302, 927, 567, 273, 81, 3289, 3234, 3054, 2730, 2282, 1757
OFFSET
1,2
COMMENTS
Principal diagonal: A213565
Antidiagonal sums: A101094
Row 1, (1,3,6,...)**(1,4,9,...): A005585
Row 2, (1,3,6,...)**(4,9,16,...): (k^5 +25*k^4 + 60*k^3 + 215*k^2 + 59*k)/60
Row 3, (1,3,6,...)**(9,16,25,...): (k^5 +35*k^4 + 30*k^3 + 505*k^2 + 149*k)/60
For a guide to related arrays, see A213500.
LINKS
FORMULA
T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
4....21....67....167...357
9....43....127...297...602
16...73....207...467...917
25...111...307...677...1302
36...157...427...927...1757
MATHEMATICA
b[n_] := n (n + 1)/2; c[n_] := n^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}] (* A213564 *)
d = Table[t[n, n], {n, 1, 40}] (* A213565 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101094 *)
CROSSREFS
Cf. A213500.
Sequence in context: A085047 A213831 A282361 * A282449 A282608 A070427
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved