Real-rooted atlas

A027467 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(16\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
264872
31536384241944
4327681228815366446656
56553603276806144051201601049760
612582912786432019660802457601536038422674816
723488102417616076855050240917504086016043008896476171136
842949672963758096384140928614429360128036700160275251211468820489795520512
97730941132877309411328338228674568455716864132120576013212057682575362949124608198359290368
101374389534720154618822656077309411328022548578304042278584320528482304044040192023592960737280102403967185807360
11241892558110723023656976384017008070492160566935683072012401718067201860257710081937768448013841203206488064018022402252878550278985728
124222124650659845805421394657283628388371660801360645639372803401614098432059528246722567441030840326643777536041523609601730150404325376491521542441841901568
1373183493944770561097752409171558475470478130544643144603255439360884419665592320176883933118464257955735797762763811454976215922769920119957094404498391041022361610649630077615917080576
1412610078956637388820491378304535756815368533728401817670439112921841664220122227880755204952750127316992825458354552832103182294319104967334009241667175972864033587986432114504499223855104229376583043016238792704
Technical details
Node id
n-798901ae44ad10ac · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8ca1e631330e3f91

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:           0;
  1:           2;
  2:          64,           8;
  3:        1536,         384,          24;
  4:       32768,       12288,        1536,         64;
  5:      655360,      327680,       61440,       5120,        160;
  6:    12582912,     7864320,     1966080,     245760,      15360,       384;
  7:   234881024,   176160768,    55050240,    9175040,     860160,     43008,     896;
  8:  4294967296,  3758096384,  1409286144,  293601280,   36700160,   2752512,  114688,   2048;
  9: 77309411328, 77309411328, 33822867456, 8455716864, 1321205760, 132120576, 8257536, 294912, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (16*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 2*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((16*n + 2*n*x)*v[i - 1])/(n - 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.