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 A213821 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 4
 2, 9, 4, 24, 16, 6, 50, 39, 23, 8, 90, 76, 54, 30, 10, 147, 130, 102, 69, 37, 12, 224, 204, 170, 128, 84, 44, 14, 324, 301, 261, 210, 154, 99, 51, 16, 450, 424, 378, 318, 250, 180, 114, 58, 18, 605, 576, 524, 455, 375, 290, 206 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Principal diagonal: A033431. Antidiagonal sums: A176060. Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002. Row 2, (2,5,8,11,…)**(2,3,4,5,…): (k^3 + 5*k^2 + 2*k)/2. Row 3, (1,2,3,4,…)**(8,11,14,17,…): (k^3 + 8*k^2 + 3*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*(2*n - (n-2)*x - (n-1)*x^2) and g(x) = (1-x)^4. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 2….9….24…50….90 4….16…39…76…130 6….23…54…102…170 8….30…69…128…210 10…37…84…154…250 12…44…99…180…290 MATHEMATICA b[n_]:=3n-1; c[n_]:=n; 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}] (* A213821 *) Table[t[n, n], {n, 1, 40}] (* A033431 *) s[n_]:=Sum[t[i, n+1-i], {i, 1, n}] Table[s[n], {n, 1, 50}] (* A176060 *) CROSSREFS Cf. A212500 Sequence in context: A200703 A080803 A228967 * A022157 A065599 A171228 Adjacent sequences:  A213818 A213819 A213820 * A213822 A213823 A213824 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jul 04 2012 STATUS approved

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Last modified April 23 07:51 EDT 2019. Contains 322381 sequences. (Running on oeis4.)