Real-rooted atlas

A027467 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118220
2324724400
35832194421688000
410497646656777657616160000
518895681049760233280259201440323200000
6340122242267481662985609331207776034566464000000
761222003247617113615872371229393280326592021772880641281280000000
81101996057697955205123809369088846526464117573120104509445806081843225625600000000
91983592903681983592903688815968460822856214528380936908842326323231352832149299241472512512000000000
1035704672266243967185807360198359290368058773123072011428107264015237476352141087744089579520373248092160102410240000000000
1164268410079232785502789857284363904388096014546347960320323252176896050283671961655870746624443418624024634368091238402027522048204800000000000
121156831381426176154244184190156894260334782873634911235104768087278087761920155161044910082011346878464191556845568133025587206569164802189721644236840964096000000000000
132082296486567116830077615917080576200517439447203848169229014515712226923028180992045384605636198467236452794368747071697715262255974809638429614080170798284851757056958464819281920000000000000
143748133675820810245830430162387927044210866228391280641871496101507235845718460310160998412707689578135552211794826302259226894581117747226147509420032193685254963210760291942443475927041207664642064384163841638400000000000000
Technical details
Node id
n-45e419df5a14be4c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e9bc54cd164ac143

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:           18,            2;
  2:          324,           72,           4;
  3:         5832,         1944,         216,           8;
  4:       104976,        46656,        7776,         576,         16;
  5:      1889568,      1049760,      233280,       25920,       1440,        32;
  6:     34012224,     22674816,     6298560,      933120,      77760,      3456,       64;
  7:    612220032,    476171136,   158723712,    29393280,    3265920,    217728,     8064,     128;
  8:  11019960576,   9795520512,  3809369088,   846526464,  117573120,  10450944,   580608,   18432,   256;
  9: 198359290368, 198359290368, 88159684608, 22856214528, 3809369088, 423263232, 31352832, 1492992, 41472, 512;
%F Formula
The row polynomials satisfy P(n,x) = (18+2*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[(18 + 2*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] = (18 + 2*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.