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 A213761 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution. 5
 1, 6, 4, 18, 15, 7, 40, 36, 24, 10, 75, 70, 54, 33, 13, 126, 120, 100, 72, 42, 16, 196, 189, 165, 130, 90, 51, 19, 288, 280, 252, 210, 160, 108, 60, 22, 405, 396, 364, 315, 255, 190, 126, 69, 25, 550, 540, 504, 448, 378, 300 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal: A172073. Antidiagonal sums: A002419. Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411. Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414. Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2. Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2. For a guide to related arrays, see A212500. LINKS Clark Kimberling, Antidiagonals n = 1..45, 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*(3*n - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1....6....18...40....75....126 4....15...36...70....120...189 7....24...54...100...165...252 10...33...72...130...210...315 13...42...90...160...255...378 MATHEMATICA b[n_]:=n; 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}] (* A213761 *) Table[t[n, n], {n, 1, 40}] (* A172073 *) s[n_]:=Sum[t[i, n+1-i], {i, 1, n}] Table[s[n], {n, 1, 50}] (* A002419 *) CROSSREFS Cf. A212500. Sequence in context: A083581 A171089 A180495 * A160248 A317858 A212891 Adjacent sequences:  A213758 A213759 A213760 * A213762 A213763 A213764 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jul 04 2012 STATUS approved

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Last modified March 20 19:57 EDT 2019. Contains 321349 sequences. (Running on oeis4.)