Real-rooted atlas

A038335 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 21\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
192130
281378441900
3729510311907926127000
4656161236214326333396194481810000
559049688905321489075014108751645408410124300000
653144174401744340101513502538023629441522054145485766121729000000
74782969781218275468527892126649735496218271569470558015403265623180108854121870000000
8430467218035387926562233468306237561848931928887016672933922419451756242812967837495237822859361656100000000
93874204898135830269759344158444134207084841446972479694337626911928652519741855565251974185556306365160824179428004658119683000000000
10348678440181358302690854262178245531540910908021704587195410607728441471481181694191750101575592255666801378643223708457148520419229016679880978201590490000000000
113138105960980544719663193968839606956577818772486530696487604937010027519284279422339754499665198389959083277533045495226382378853538517607518355165130821684189935027750054222117714700000000000
122824295364817908027021468101486346775506789338252698380414402582666649515471029752888248421155809939735648423116198794712812283711123242289512738663387066078089170643709462546378299700585598687355827511386641531441000000000000
1325418658283297710326345931310794456884303829235257556571046538723357466644352262638101359906277039318537564152841642507658764968996287438840283869574337260590407168278153477655104669039294221305324842575227886063181883223699715447237773911946115943230000000000000
1422876792454961747308553528726113341797285190111057856774661774366787914304079718813167693341903868778110869266966635407232956513785776944192860362156459612610603938966878260640609381095461357970747377619294823643388159567654219804586430930811194635195951290520863243919932521508681478296900000000000000
Technical details
Node id
n-846c10917dd517bf · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5184fe74b2189306

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:         9,         21;
  2:        81,        378,         441;
  3:       729,       5103,       11907,         9261;
  4:      6561,      61236,      214326,       333396,        194481;
  5:     59049,     688905,     3214890,      7501410,       8751645,       4084101;
  6:    531441,    7440174,    43401015,    135025380,     236294415,     220541454,      85766121;
  7:   4782969,   78121827,   546852789,   2126649735,    4962182715,    6947055801,    5403265623,    1801088541;
  8:  43046721,  803538792,  6562233468,  30623756184,   89319288870,  166729339224,  194517562428,  129678374952,   37822859361;
  9: 387420489, 8135830269, 75934415844, 413420708484, 1446972479694, 3376269119286, 5251974185556, 5251974185556, 3063651608241, 794280046581;
%F Formula
The row polynomials satisfy P(n,x) = (9+21*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[(9 + 21*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] = (9 + 21*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.