This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A138094 A triangular sequence of eight back recursive polynomials that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]:k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}]. 1
 1, 0, 2, 2, 0, 4, 0, 10, 0, 12, 24, 0, 36, 0, 32, 0, 148, 0, 140, 0, 86, 432, 0, 656, 0, 512, 0, 232, 0, 3076, 0, 2976, 0, 1782, 0, 624, 10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680, 0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512, 188320, 0, 459736, 0, 382592, 0, 198688 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums are: {1, 2, 6, 22, 92, 374, 1832, 8458, 46272, 236048, 1306268}; The alternating orthogonal integration is: Table[Integrate[P[x, n]*P[x, m]*Exp[ -x^2/2], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0,10}] // TableForm; This sequence is the result of a thought experiment for 8th derivatives. The lower 7 row sums are the same as k=6: only the higher values are really different. LINKS FORMULA k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}]; out_n,m=Coefficients(P(x,n)). EXAMPLE {1}, {0, 2}, {2, 0, 4}, {0, 10, 0, 12}, {24, 0, 36, 0, 32}, {0, 148, 0, 140, 0, 86}, {432, 0, 656, 0, 512, 0, 232}, {0, 3076, 0, 2976, 0, 1782, 0, 624}, {10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680}, {0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512}, {188320, 0, 459736, 0, 382592, 0, 198688, 0, 64800, 0, 12132} MATHEMATICA Clear[P, x]:k=8; Table[P[x, -n] = 0, {n, 1, k}]; P[x, 0] = 1; P[x_, n_] := P[x, n] = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P[x, n - m], n^(m/2)*P[x, n - m]], {m, 1, k}];; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}]; CROSSREFS Similar to but different from A138093. Sequence in context: A138092 A138090 A138093 * A060821 A191718 A286777 Adjacent sequences:  A138091 A138092 A138093 * A138095 A138096 A138097 KEYWORD nonn,uned,tabl AUTHOR Roger L. Bagula, May 02 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)