Real-rooted atlas

A038279 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112315
2144819234
317281647405273807
420736298891223117828164719
52488325105433098257411572902431150848
629859848404641710847924858903973052843172921411459
735831808135027567153166545735291271699299019595521071632187416616939
8429981696213290892931620805112003611302628419141103886874909354639366065618470382220
9515978035233283579023633161759855156207887521092893290469020302158446700240271418141717619683179730885825
1061917364224514733026401123954190479912727521843306609156755851899141614373160461912330823578001718325905019165590493974638204503
11743008370688790674748868723855953365585304464215853271968023276703071104293855541517881058001196531664769154330466407112452051753755317714791473367692186
12891610044825612083023744236945312740304659171094271338862256366485116470125102229171104866867719896692110391881113801145184673347740377029152870312841605842745314412187632659014477
131069932053790721839201862980303851878396716574516289179190300835156607845681186908468596429692399275909847203390456489636736561674344212574632632302443372634266624806711337230853207261990159432354288555512380155
141283918464548864279090075608417611588557169979243193676291148978147704247263672057510652756344573060508431081787434500943172691887553916704043710807547836004696305906392103336105571999215381741325864396026807970309644347829691396011856665615858
Technical details
Node id
n-8ad1310c95b3d356 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d99d39db05a8c355

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,           3;
  2:        144,          81,           9;
  3:       1728,        1647,         405,          27;
  4:      20736,       29889,       12231,        1782,          81;
  5:     248832,      510543,      309825,       74115,        7290,        243;
  6:    2985984,     8404641,     7108479,     2485890,      397305,      28431,       729;
  7:   35831808,   135027567,   153166545,    73529127,    16992990,    1959552,    107163,     2187;
  8:  429981696,  2132908929,  3162080511,  2003611302,   628419141,  103886874,   9093546,   393660,    6561;
  9: 5159780352, 33283579023, 63316175985, 51562078875, 21092893290, 4690203021, 584467002, 40271418, 1417176, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (12+3*x)*P(n-1,x) + 3*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 + 3*x)*P[n - 1, x] + 3*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 + 3*x)*v[i - 1] + 3*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.