Real-rooted atlas

A038279 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + n x\right)\,P_{n}(x) = \left(27 + 18\,x + 3\,x^{2} + 9\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
28110827216
372914587291083024
465611749613122388840541472
55904919683019683087480182251458559872
65314412125764265720515746404920757873251037464960
747829692232052233480783248005801033357524800583214891749698537472
84304672122958251240176939635712835218600435059521392115736041259712590491289945088
93874204892324522934464904586848212327523013270470120530818831248730851018336478296919683016769286144
10348678440123245229340523017660156198727824045199057050216955473847030964430153055008021523360517714700649539216710774784
11313810596092301277704665753194261657670925682206392438068503579765318361392130957143788111448071027089658768776506430436121257642786281390080
12282429536481225943629184862134498025829205110818640862979139247555230664911842505835722852818232072768191773142055315675954003472435494229582512690873335664401793024
132541865828329220295038455186608851153655410769979657808811218728810217580774847433566418832542248121595552541897644874915240879233521654501354249842661343057695280832176122320522454721122861056
142287679245496121351672957963669392937113381712336522153490081413559830087405113084786406992465966125404078928719945754156894237322005827232684745693404265887004379564084231984509242709432812385772717445355777633090469888
Technical details
Node id
n-bbb58b7a94b80108 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-20d8219123bfe5ea

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,          6;
  2:        81,        108,         27;
  3:       729,       1458,        729,        108;
  4:      6561,      17496,      13122,       3888,        405;
  5:     59049,     196830,     196830,      87480,      18225,       1458;
  6:    531441,    2125764,    2657205,    1574640,     492075,      78732,      5103;
  7:   4782969,   22320522,   33480783,   24800580,   10333575,    2480058,    321489,    17496;
  8:  43046721,  229582512,  401769396,  357128352,  186004350,   59521392,  11573604,  1259712,   59049;
  9: 387420489, 2324522934, 4649045868, 4821232752, 3013270470, 1205308188, 312487308, 51018336, 4782969, 196830;
%F Formula
The row polynomials satisfy (3+n*x)*P(n,x) = (27+(9*n+18)*x+(3*n+3)*x^2)*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[Together[((27 + (9*n + 18)*x + (3*n + 3)*x^2)*P[n - 1, x])/(3 + n*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] = ((27 + (9*n + 18)*x + (3*n + 3)*x^2)*v[i - 1])/(3 + n*x)); 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.