Real-rooted atlas

A357367 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
218421272
326412248401202448
454004620058200193201680130800
5141840215784044856002610720514080302409940320
64551120120692880391466880356680800119599200156340806652801009290240
71725292807887841920386045049605181841728026358393600575068032053621568017297280131145880320
8754536384059097076416042713178681608136500803200589794670080019113840345602931370041602049727680051891840021129818734080
9373944211200499921232832005259477484800001386844870118400137789648156160062258376660480013970321164800015889281408000864518054400176432256004120113588595200
1020711190931200471618393079680071509045803148800256485567168230400340208633974809600205985345927961600634501474903296001043932015526400091584219767040039891333081600670442572800953771359614796800
1112677845517568004911199625477376001065784063105681920051341854476386073600892241765175720960007046112627238164480028624350370475059200639017045116369920080378689610419200056055703511808000199925975208960028158588057600258053776484091033600
1284991791159475200559711717106750208001729922774380965964800110872462775803432704002489461692878252252160025147470633739072819200131089986686446115328003814024045028227430400645092813287341619200640443150570263040003636090317289830400108157136729241600129529505064960080551218162751015680000
13619309114605941760069304158987480248832003040109728919616697344002573784665984502325555200738762291205219804723200094075828613473742171136006169154750325705032294400227549635819471082557440049700534703675574118400065869890103703702016000529268179114841886720024939869964217528320062821809956505600006476475253248000028693012794363344422195200
1448736176191602099200092653741640547716428800057541689309017146641408000639983608995766497606144000232892608119863048182425600036967684465172788805237760003004314836618336451065856000137694642888939559847424000037773217077973780641638400064175756983855228647936000684311859472986333388800045413076858151242240000018047022116938214400000389948574998062080000349729663675392000011554631733901485433171968000
Technical details
Node id
n-4219079737abb529 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b832379b06e619bb

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:            2,              2;
  2:           18,             42,              12;
  3:          264,           1224,             840,              120;
  4:         5400,          46200,           58200,            19320,             1680;
  5:       141840,        2157840,         4485600,          2610720,           514080,           30240;
  6:      4551120,      120692880,       391466880,        356680800,        119599200,        15634080,          665280;
  7:    172529280,     7887841920,     38604504960,      51818417280,      26358393600,      5750680320,       536215680,       17297280;
  8:   7545363840,   590970764160,   4271317868160,    8136500803200,    5897946700800,   1911384034560,    293137004160,    20497276800,    518918400;
  9: 373944211200, 49992123283200, 525947748480000, 1386844870118400, 1377896481561600, 622583766604800, 139703211648000, 15889281408000, 864518054400, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n-3+(4*n-2)*x)*P(n-1,x) + (4*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n)*P[n - 2, x] + (6*n - 3 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], 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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n)*v[i - 2] + (6*n - 3 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.