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 A173568 A symmetrical triangle based on the transpose and antidiagonal of: t(n,q) = (Sum[(1 + (-1)^n)*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4. 0
 6, 19, 19, 68, 56, 68, 261, 211, 211, 261, 1030, 1044, 654, 1044, 1030, 4103, 5819, 2993, 2993, 5819, 4103, 16392, 33560, 19102, 9840, 19102, 33560, 16392, 65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545, 262154, 1136836, 1019606 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Row sums are {6, 38, 192, 944, 4802, 25830, 147948, 902504, 5844862, 40010854, ...}. LINKS FORMULA t(n,q) = (Sum[(1 + (-1)^n)*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4;even n only: t1(n,q) = (t(n,q) + Transpose(t(n,q))); out_n,m = antidiagonal(t1(n,q)) EXAMPLE {6}, {19, 19}, {68, 56, 68}, {261, 211, 211, 261}, {1030, 1044, 654, 1044, 1030}, {4103, 5819, 2993, 2993, 5819, 4103}, {16392, 33560, 19102, 9840, 19102, 33560, 16392}, {65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545}, {262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154}, {1048587, 6625283, 7600477, 3608745, 1122335, 1122335, 3608745, 7600477, 6625283, 1048587} MATHEMATICA Clear[t, a, b, n, q]; t[n_, q_] := (Sum[(1 + (-1)^n)*( 1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4; a = Table[FullSimplify[ExpandAll[t[n, q]]], {q, 1, 20}, {n, 2, 40, 2}]; b = (a + Transpose[a]); Table[Table[b[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]; Flatten[%] CROSSREFS Sequence in context: A119986 A245869 A184197 * A012589 A009048 A235537 Adjacent sequences:  A173565 A173566 A173567 * A173569 A173570 A173571 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Feb 22 2010 STATUS approved

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