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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 (list; table; graph; refs; listen; history; text; internal format)
 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 Clark Kimberling, Antidiagonals n=1..60, 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*((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 Cf. A212500 Sequence in context: A084160 A238139 A268722 * A324003 A176466 A128155 Adjacent sequences:  A213825 A213826 A213827 * A213829 A213830 A213831 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jul 04 2012 STATUS approved

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Last modified August 6 12:33 EDT 2020. Contains 336246 sequences. (Running on oeis4.)