Real-rooted atlas

A114687 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(49 + 168\,x + 144\,x^{2} - 49\,n - 168\,n x - 144\,n x^{2}\right)\,P_{n-2}(x) + \left(9 + 12\,x + 18\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191221
289228144461
394535284320172810521
41057751576927367257620736248201
51231297393801746720202176011404802488326020301
6147684110524348307562404702752039502080171694082985984149442421
7181304011494585125201400969878803201101703680717631488250822656358318083781598961
82267394892121913776856562918419434594816270232381442347978752012373917696358318080042998169697238983121
92878666857301444421401384423228803652532352006084800870406614820495364670317854722051012689925030785843251597803522534281496901
103700632677742867746082022068639972006637369910400128869284096001682967767040014915169239040881139990528032958096998406965703475206191736422466807390331101
114807509909936103311047064348085496404801175434500240002606438964096003975537007411204243141789286403156076398182401594178637004805164940132352095352740904967430083706881778401025085321
1263016116314738700181711581654447166568044820396420715052805085737508614400887449412044800011084527523266560995251327893504063475553393049602787364943953920792789931524096129283456499712891610044825647740599368927961
1383240669582937124168152228166884595175689302723481371579184012896420191043002112189463084748061696271167908818599936285458678167044096220256842860527616122454089900752896473698794995712001195797671785267217386395874099201069932053790721290992440100836221
14110698050949364117741484564029100130712638665707280586190401038889920178546243265470080039015986356757329926296546117904666624760685845046855270468935461695526666244646261349440815104228139537598093721678587475261810278417768896583329382423217525567258624128391846454886435135485198939396101
Technical details
Node id
n-3a09f72d6a4c83d4 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-950a351d3feace76

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,          12;
  2:         89,         228,          144;
  3:        945,        3528,         4320,         1728;
  4:      10577,       51576,        92736,        72576,        20736;
  5:     123129,      739380,      1746720,      2021760,      1140480,       248832;
  6:    1476841,    10524348,     30756240,     47027520,     39502080,     17169408,      2985984;
  7:   18130401,   149458512,    520140096,    987880320,   1101703680,    717631488,    250822656,     35831808;
  8:  226739489,  2121913776,   8565629184,  19434594816,  27023238144,  23479787520,  12373917696,   3583180800,   429981696;
  9: 2878666857, 30144442140, 138442322880, 365253235200, 608480087040, 661482049536, 467031785472, 205101268992, 50307858432, 5159780352;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(49*n-49+(168*n-168)*x+(144*n-144)*x^2)*P(n-2,x) + (18*n+9+(24*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(49*n - 49 + (168*n - 168)*x + (144*n - 144)*x^2)*P[n - 2, x] + (18*n + 9 + (24*n + 12)*x)*P[n - 1, x])/(n + 2)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(49*n - 49 + (168*n - 168)*x + (144*n - 144)*x^2)*v[i - 2] + (18*n + 9 + (24*n + 12)*x)*v[i - 1])/(n + 2)); 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. [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