Real-rooted atlas

A038327 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116622
225619236484
340964608172821610648
4655369830455296138241296234256
510485761966080147456055296010368077765153632
616777216377487363538944017694720497664074649646656113379904
72684354567046430727927234564954521601857945604180377652254722799362494357888
842949672961288490188816911433728126835752965945425920178362777633443020835831808167961654875873536
96871947673623192823398434789235097630440580710417122826649664210599936160526499842579890176241864704100776961207269217792
101099511627776412316860416069578470195206957847019520456608710656020547391979526421059993601375941427201934917632016124313606046617626559922791424
11175921860444167256776743321613606456393728015307263442944011480447582208060272349806592226021311774726054142279680113515167744014189395968010642046976362797056584318301411328
1228147497671065612666373951979522612439627595776326554953449472027553074197299201653184451837952723268197679104232479063539712544872805171209081213419520102163650969669657034752217678233612855002631049216
1345035996273704962195504818343116849398858412720128679234303174901766367821592264704042982795747786752214913978738933768059274202710016226667086951219247222309781504070833464672256724433161420845277072588813060694016282810057883082752
147205759403792793637830236869912166492211202370410905613831680355561635841426392036667293696106979402750047027260175914046901452825789677448672051284622379128455168211555947821137923966674021646336540910093860864507103212994562925595459584783641640966221821273427820544
Technical details
Node id
n-a5bbe7b7541d0f7e · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-10cba46fcb583bff

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:          16,            6;
  2:         256,          192,           36;
  3:        4096,         4608,         1728,          216;
  4:       65536,        98304,        55296,        13824,         1296;
  5:     1048576,      1966080,      1474560,       552960,       103680,        7776;
  6:    16777216,     37748736,     35389440,     17694720,      4976640,      746496,       46656;
  7:   268435456,    704643072,    792723456,    495452160,    185794560,    41803776,     5225472,     279936;
  8:  4294967296,  12884901888,  16911433728,  12683575296,   5945425920,  1783627776,   334430208,   35831808,   1679616;
  9: 68719476736, 231928233984, 347892350976, 304405807104, 171228266496, 64210599936, 16052649984, 2579890176, 241864704, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (16+6*x)*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[(16 + 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; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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

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.