Real-rooted atlas

A038277 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111718
212115449324
31331254116173435832
4146413726835574150922401104976
5161051512435652190415030132055168071889568
6177156167641421076113591306604357815110926211764934012224
71948717186806489165721479175765205111850585427065879058973823543612220032
82143588811091281576243058169230934676082460712870125272655239859481272471784576480111019960576
92357947691135046095033437536964451042215532487221148263100498216213153628796358735330857071529940353607198359290368
10259374246011650563383704726613326058020919583608932387718106821096075643617247918901315362879603138934144544388967702824752493570467226624
112853116706111997181694277635466902724512131640870195154402701984301375587708587487537399637383978972710790126603677161526855325458534179505129197732674364268410079232
123138428376721239661803313248388163115963417793073276286025476445827409525939653933362419258227920223610504487956485641779213463295118163432017402255847338514261007130076138412872011156831381426176
133452271214393128559698228161110904612050752422544409478508898404792417035506546367131405885293934180846559964250356962962906811948855050502374224342694622051075287231358341866200980043419793040697439688901040720822964865671168
1437974983358324133832257901052381399425213179893935621732699124572623380322234680017933931373895927475732981296279307550785318518394963066872796295608313010975499435914413985584072960995798316975561215240641337021114920907602678678223072849374813367582081024
Technical details
Node id
n-82e1f9a697cd4770 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-428cb055b0457fc7

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:         11,           7;
  2:        121,         154,          49;
  3:       1331,        2541,        1617,         343;
  4:      14641,       37268,       35574,       15092,        2401;
  5:     161051,      512435,      652190,      415030,      132055,       16807;
  6:    1771561,     6764142,    10761135,     9130660,     4357815,     1109262,      117649;
  7:   19487171,    86806489,   165721479,   175765205,   111850585,    42706587,     9058973,     823543;
  8:  214358881,  1091281576,  2430581692,  3093467608,  2460712870,  1252726552,   398594812,   72471784,   5764801;
  9: 2357947691, 13504609503, 34375369644, 51042215532, 48722114826, 31004982162, 13153628796, 3587353308, 570715299, 40353607;
%F Formula
The row polynomials satisfy P(n,x) = (11+7*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[(11 + 7*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] = (11 + 7*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.