Real-rooted atlas

A013612 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 330\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0121330225676
1133154457425337517576
2146417986016335014850050625456976
3161051109807529947504083750278437575937511881376
41771561144945904941337589842500918843755011875011390625308915776
5194871711860139057609659751729468125235836562519295718758770781251708593758031810176
621435888123384605201116083430030438639000518840437505660077500038591437500150356250002562890625208827064576
7235794769128938448935157846085100502237543500102730406625014008691812501273517437500744263437500253726171875384433593755429503678976

All 15 rows

Technical details
Node id
n-7e240212e767dd6b · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c62c9860d860e562

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          121,           330,            225;
  1:         1331,          5445,           7425,            3375;
  2:        14641,         79860,         163350,          148500,           50625;
  3:       161051,       1098075,        2994750,         4083750,         2784375,          759375;
  4:      1771561,      14494590,       49413375,        89842500,        91884375,        50118750,        11390625;
  5:     19487171,     186013905,      760965975,      1729468125,      2358365625,      1929571875,       877078125,       170859375;
  6:    214358881,    2338460520,    11160834300,     30438639000,     51884043750,     56600775000,     38591437500,     15035625000,      2562890625;
  7:   2357947691,   28938448935,   157846085100,    502237543500,   1027304066250,   1400869181250,   1273517437500,    744263437500,    253726171875,     38443359375;
  8:  25937424601,  353692153650,  2170383670125,   7892304255000,  18833907881250,  30819121987500,  35021729531250,  27289659375000,  13954939453125,   4228769531250,   576650390625;
  9: 285311670611, 4279675059165, 29179602676125, 119371101856875, 325557550518750, 621518960081250, 847525854656250, 825512196093750, 562849224609375, 255840556640625, 69774697265625, 8649755859375;
%F Formula
The row polynomials satisfy P(n,x) = (11+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 330*x + 225*x^2.
%t Mathematica
P[0, x_] := 121 + 330*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 15*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] = 121 + 330*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 15*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.