Real-rooted atlas

A038303 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(14 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114923
219625281529
327445292340272912167
4384169878495256408246561279841
5537824172872022226401428840459270590496436343
67529536290424964667544040007520192893404960116531441148035889
71054135044743607689148386249801842406301184402430456845208121847829693404825447
814757890567589772288170769876482195612697617643316320907370553629165482085356925284304672178310985281
9206610467841195389135363073857776644610786664964446115712642858217243841224950247363374862926454238868463874204891801152661463
102892546549761859494210560537925110912092215733299201037426999616080030082827524287325865760157493603232037967207922054238868460348678440141426511213649
11404956516966428636210842624920449634227201775152866009602282339399155202054105459239681320496366654086063503724432019489833399960417639287142053696479775431381059609952809757913927
125669391237529643735303832371215463553855016963313618683217920479291273822592049298531021752323697389826631424203733725140915281857300279832023387800079952045105043011336527201801431228242953648121914624432020321
137937147732541446633187747909632255851527419371526030786003456614496923346484124160112154158074486528961321354924170246179922995941094429796057301858848106414490363781602736372609354384479753639302392514021756395422541865828329504036361936467383
1411112006825558016100008061430022144417890828118306816107457641516136038418996975910888335362442468331399928832235523731956421708817303784388635064329733378718607223684171448022260238721340822578583648163134390443442294450374132126751164982057023524842287679245496111592836324538749809
Technical details
Node id
n-6d0748781e5e30ed · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9111d60e4a92b32c

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:          14,            9;
  2:         196,          252,           81;
  3:        2744,         5292,         3402,          729;
  4:       38416,        98784,        95256,        40824,         6561;
  5:      537824,      1728720,      2222640,      1428840,       459270,        59049;
  6:     7529536,     29042496,     46675440,     40007520,     19289340,      4960116,       531441;
  7:   105413504,    474360768,    914838624,    980184240,    630118440,    243045684,     52081218,     4782969;
  8:  1475789056,   7589772288,  17076987648,  21956126976,  17643316320,   9073705536,   2916548208,   535692528,   43046721;
  9: 20661046784, 119538913536, 307385777664, 461078666496, 444611571264, 285821724384, 122495024736, 33748629264, 5423886846, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (14+9*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[(14 + 9*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] = (14 + 9*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.