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 A123243 An even-odd switched polynomial recursion between a Bessel-like polynomial and a Mohanmian-like polynomial ( A008302) to give a new triangular array: odd:p(k, x)=2*x*p(k - 1, x) + (1 - x2)*p(k - 2, x) even:p(k, x)=Sum[x^m, {m, 0, k}]*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 COMMENTS Polynomials here go up as the square powers ( 1,4,9,16 ) in jumps as pairs of two. Row sums appear to be new: Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[ n, x], x]]}], {n, 0, 15}] {1, 2, 6, 22, 110, 858, 6006, 71214, 640926, 10183602, 112019622, 2230208838, 28992714894, 693594948618, 10403924229270, 290616283470942} The Polynomial Roots all contain -2: the polynomials seem to represent a new kind of differential structure. 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 B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4. LINKS FORMULA odd:p(k, x)=2*x*p(k - 1, x) + (1 - x2)*p(k - 2, x) even:p(k, x)=Sum[x^m, {m, 0, k}]*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: A104345 A244516 A002339 * A037193 A291615 A003986 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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