Real-rooted atlas

A033282 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(6 + 3\,n\right)\,P_{n}(x) = \left(18 - 4\,x^{2} + 36\,n + 27\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-18\,x - 18\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
2454511101
3378567279451269
4340268045040163819717081
5320768019079380388809420903240849
631274193822311627557618052782355370342793511741
73127410109459351629728113378365653845519022853050252079352515549
83189958212759832822183761621891870013410797452216164126181441730124103049801029681
93305956681487680506295796124034103553122512390608122634275439658928481931824981131451885912414177369
1034712545141735627257038844990990512423285404411808856025902460584105019059602903215770523697265556598252646723194922521301
113684870176420266785970250422196979074898991629073803363678050650647062424703145359285616296920206635552203307449915315974043136488693094216933509
123948075189002368845113400648537151046410712444012820118894281669909340344838440532552888296022203483207846718024704601438534825202069244144017952232507103937349575333021801
13426392120412027715487826780828042969639601505532672072001858496442968881644794444238661073631471541205233241265595219047722450328511285791416498375632898412848163494410208651544372693519800645021406369
144637014309480532459100166363510508789134320752085590458965195283447890781555527905675382134552052271043970147114531307010277348737448324665115737028762955537953675478719662953186431979282536687958103965899196880151913020241953611181
Technical details
Node id
n-a84321342ff00533 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6a81dd80276de4b8

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:         6,          3;
  2:        45,         45,         11;
  3:       378,        567,        279,         45;
  4:      3402,       6804,       5040,       1638,        197;
  5:     32076,      80190,      79380,      38880,       9420,        903;
  6:    312741,     938223,    1162755,     761805,     278235,      53703,      4279;
  7:   3127410,   10945935,   16297281,   13378365,    6538455,    1902285,    305025,    20793;
  8:  31899582,  127598328,  221837616,  218918700,  134107974,   52216164,  12618144,  1730124,  103049;
  9: 330595668, 1487680506, 2957961240, 3410355312, 2512390608, 1226342754, 396589284, 81931824, 9811314, 518859;
%F Formula
The row polynomials satisfy (3*n+6)*P(n,x) = (36*n+18+27*n*x+(4*n-4)*x^2)*P(n-1,x) - (18*x+18*x^2+4*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n + 18 + 27*n*x + (4*n - 4)*x^2)*P[n - 1, x] - (18*x + 18*x^2 + 4*x^3)*D[P[n - 1, x], x])/(3*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((36*n + 18 + 27*n*x + (4*n - 4)*x^2)*v[i - 1] - (18*x + 18*x^2 + 4*x^3)*deriv(v[i - 1], x))/(3*n + 6)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023