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 A123243 Triangular array: odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x), even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x). 0
 1, 1, 1, 2, 1, 2, 3, 3, 3, 1, 12, 16, 17, 18, 6, 12, 28, 45, 63, 69, 69, 57, 41, 24, 6, 120, 268, 434, 613, 672, 684, 570, 410, 240, 60, 120, 388, 822, 1435, 2107, 2791, 3361, 3771, 3891, 3683, 3249, 2636, 1964, 1280, 710, 300, 60, 1680, 5312, 11240, 19656, 28885 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 REFERENCES E. S. R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966, pages 36-40 S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 110 LINKS Table of n, a(n) for n=1..57. B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4. FORMULA odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x); even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x) EXAMPLE {1}, {1, 1}, {2, 1}, {2, 3, 3, 3, 1}, {12, 16, 17, 18, 6}, {12, 28, 45, 63, 69, 69, 57, 41, 24, 6}, {120, 268, 434, 613, 672, 684, 570, 410, 240, 60}, {120, 388, 822, 1435, 2107, 2791, 3361, 3771, 3891, 3683, 3249, 2636, 1964, 1280, 710, 300, 60} MATHEMATICA p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = If[Mod[k, 2] == 0, 2*(k - 1)*p[k - 1, x] - x*p[k - 2, x], Sum[x^m, {m, 0, k}]*p[k - 1, x]]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w] CROSSREFS Cf. A008302. Sequence in context: A244516 A363264 A002339 * A037193 A291615 A361789 Adjacent sequences: A123240 A123241 A123242 * A123244 A123245 A123246 KEYWORD nonn,uned,tabf AUTHOR Roger L. Bagula, Oct 07 2006 STATUS approved

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Last modified February 28 07:05 EST 2024. Contains 370387 sequences. (Running on oeis4.)