Real-rooted atlas

A038208 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
214425664464
317286272384051212352
420736139264157696491524096370944
52488322951168552960030310405734403276812366848
6298598461014016178438144153354240494796806291456262144451825664
735831808124416819254843801606948257792340393984072299315266060288209715217907728384
84299816962517001830416351599001629401232179220535941529664827162624974756249667108864016777216763750318080
951597803525068402196484779807866880118925715046401138791284736050138877788161103471050752123614527488664377753613421772834820043571200
1061917364224101780826357761378893420298244663710371020805962087373209603518254672773121063193652756481723355627520014938433126406442450944010737418241687646846844928
1174300837068820405699162931239423262379212801789247207191347229964152738938880230645941969879049193765654757376202243674891878425140054196224017362405294080614180323328858993459286553933366951936
1289161004482564087083899551744112017590594830336675667604492255232146156249708861849614390721201548820487361426928613457922110758241645363203528593518048051234696463804006401954038320988165772436045824687194767364678591155904970752
131069932053790728181300679462092831691892078516633602522017448647983104069714090707837255680865242502247628144645568113853291980390420242297586079236096437033766703280947257373777739749785645734323596296192214047426137292853601191854080549755813888265625065139166248960
141283918464548864163711608153545113689391801874203541504933279795176087224320326918138703667789824050569737373503643975684033062313773290749952182192534411674648576049408404338312740864083156674637652295680879745999003923251257992201490766233622912722811224064492581209243648439804651110415792094289055163875328
Technical details
Node id
n-3408a1ca2fd4b6f2 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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:         12,            8;
  2:        144,          256,            64;
  3:       1728,         6272,          3840,            512;
  4:      20736,       139264,        157696,          49152,           4096;
  5:     248832,      2951168,       5529600,        3031040,         573440,         32768;
  6:    2985984,     61014016,     178438144,      153354240,       49479680,       6291456,        262144;
  7:   35831808,   1244168192,    5484380160,     6948257792,     3403939840,     722993152,      66060288,      2097152;
  8:  429981696,  25170018304,  163515990016,   294012321792,   205359415296,   64827162624,    9747562496,    671088640,   16777216;
  9: 5159780352, 506840219648, 4779807866880, 11892571504640, 11387912847360, 5013887778816, 1103471050752, 123614527488, 6643777536, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (12+8*x)*P(n-1,x) + 8*x*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[(12 + 8*x)*P[n - 1, x] + 8*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; for(n = 1, N, my(i = n + 1); v[i] = (12 + 8*x)*v[i - 1] + 8*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.