Real-rooted atlas

A116666 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-4 - 12\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14913
2124845105
332180324189725
480576151217287294625
51921680576097208100267328125
64484608194404320053460349929477165625
71024120966048016632027216026535614288432805953125
82304307201774085806081179360152409612247205598721115375390625
951207603249766418869764572288734832078382085353776212576437397730078125
101126418432013478405806080163296003135283241640480377913602243862078732001240029166015625
11245764392963548160171072005474304012214540819399564821939984017321040090935460285797164074381908203125
1253248103219291238404866048017449344044341862481909273611085465601091225520762125760358545528102036672132860254931640625
1311468823961602300313613442457653374464015211722243202467840504388684859445809285178990960324249868813817466003592541164304672126611328125
142457605505024570286083622993921577511936498161664011769069312211362877442900234937630263321088236432196001341215366452230021481249949232138706101142822265625
Technical details
Node id
n-334042a33192acbb · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-7788c6846a0d7f3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    4,     9;
  2:   12,    48,     45;
  3:   32,   180,    324,     189;
  4:   80,   576,   1512,    1728,     729;
  5:  192,  1680,   5760,    9720,    8100,    2673;
  6:  448,  4608,  19440,   43200,   53460,   34992,    9477;
  7: 1024, 12096,  60480,  166320,  272160,  265356,  142884,   32805;
  8: 2304, 30720, 177408,  580608, 1179360, 1524096, 1224720,  559872,  111537;
  9: 5120, 76032, 497664, 1886976, 4572288, 7348320, 7838208, 5353776, 2125764, 373977;
%F Formula
The row polynomials satisfy P(n,x) = -(4+12*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 12*x + 9*x^2)*P[n - 2, x] + (4 + 6*x)*P[n - 1, 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] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 12*x + 9*x^2)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.