Real-rooted atlas

A027467 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
232234
3768963867
4163843072192419652
53276808192076803205417605
6629145619660802457601536048068519142
7117440512440401926881280573440268806727168962983
821474836489395240961761607681835008011468804300889683282709384
938654705664193273528324227858432528482304412876802064384645121152962781816969
1068719476736038654705664096636764160140928614401321205760825753603440640921601440101185878764970
111209462790553675591424409602126008811520354334801920387553689602906652672151388160540672012672017601122175932904939
1221110623253299214513553486643245354854645760850403524608010630044057609301288550458133053442595225608110080168960211212411262755691596
133659174697238528274438102292889694338097663180819653770346496027638114549760276381145497620152791859210796138496421724160117145602196482496137574089083986893
1463050394783186944512284457613393921921066716050227244024445576151046878819621273607738672073932864488933949444030558371841889324236865601536016400384279552291214138664092460683118
Technical details
Node id
n-1c27287a9c35f694 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d75da96e6240dd31

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:           1;
  2:          32,           2;
  3:         768,          96,          3;
  4:       16384,        3072,        192,         4;
  5:      327680,       81920,       7680,       320,        5;
  6:     6291456,     1966080,     245760,     15360,      480,       6;
  7:   117440512,    44040192,    6881280,    573440,    26880,     672,     7;
  8:  2147483648,   939524096,  176160768,  18350080,  1146880,   43008,   896,    8;
  9: 38654705664, 19327352832, 4227858432, 528482304, 41287680, 2064384, 64512, 1152, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (16*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n + 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.