Real-rooted atlas

A232535 → basis A013610 → basis A013610

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-9 - 54\,x - 81\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191827
281288243612
372936905913291613248
46561430561012509914432805282816
5590494760821475982216950414762253542946011136
65314415081904195615813841538439963051203128563720087127586304
7478296952911594243314847598471092841113639663828858264126177382637522706812928
84304672154077443228920333968550680112151650297901632586752010222799076329068267238742048957418334208
93874204895448100626332197935481147523215682459665755903363584567402897395074361485398852643964603004138742048901217932296192
103486784401542630822403715468424551468550846160369273084489061325334226086749877844926479666742321620609478613174649045868003835462841125833864167424
11313810596095354393295784067647269525181054157293805226009585741010219013019457213719007520253012523664807596875102624054117275386231990985331293349129376572715308547965946036224
122824295364815242598270928437578230681182165960113758007058668241674351589030142412608252030619166634028173892167390242180977228864071112322807789227235662133496547860000585972408367158397425311622970048905216
1325418658283295099618318085046388058450661825276286491695369179883705504015233876092370088424281955900831047656726609178003696539716794859488873604231309660839016165938910438746449728241468500866455669933979335586121596606246536065140129792
1422876792454961493280837310096485693963562334528889200004808744115702783039774125329151642360965928683506960176423789104785367509357070411848831457748979239758230120976126405698979610844767392251806928635859925543724642724604772595688057076243431518868244155229302136881283072
Technical details
Node id
n-4182d3b7552bf009 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-e01ea07d291a6f88

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         9,         18;
  2:        81,        288,         243;
  3:       729,       3690,        5913,         2916;
  4:      6561,      43056,      101250,        99144,        32805;
  5:     59049,     476082,     1475982,      2169504,      1476225,       354294;
  6:    531441,    5081904,    19561581,     38415384,     39963051,     20312856,      3720087;
  7:   4782969,   52911594,   243314847,    598471092,    841113639,    663828858,    264126177,     38263752;
  8:  43046721,  540774432,  2892033396,   8550680112,  15165029790,  16325867520,  10222799076,   3290682672,   387420489;
  9: 387420489, 5448100626, 33219793548, 114752321568, 245966575590, 336358456740, 289739507436, 148539885264, 39646030041, 3874204890;
%F Formula
The row polynomials satisfy P(n,x) = -(9+54*x+81*x^2)*P(n-2,x) + (10+18*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 18*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 18*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 54*x + 81*x^2)*P[n - 2, x] + (10 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 54*x + 81*x^2)*v[i - 2] + (10 + 18*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023