Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 3\,x + 6\,n x\right)\,P_{n}(x) = \left(1 + 9\,x + 27\,x^{2} + 27\,x^{3} + 6\,n x + 36\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11122740
21241624324051024
3136405216060758748510322528
41487566048283508164814288413996859049458752
51601215129608505036741610716302099520265720519683006495398912896
61721782237602004751154736471517213856832292292554330260042869574255091686908733167772160
718424573931240540529189161532430960046272177324147394053660650188539773778096628694703312487308717445353087007744

All 15 rows

Technical details
Node id
n-7076c2900694d81e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f47695d0d66fbaa8

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  12,   27;
  2: 1,  24,  162,   432,     405;
  3: 1,  36,  405,  2160,    6075,     8748,     5103;
  4: 1,  48,  756,  6048,   28350,    81648,   142884,    139968,      59049;
  5: 1,  60, 1215, 12960,   85050,   367416,  1071630,   2099520,    2657205,    1968300,      649539;
  6: 1,  72, 1782, 23760,  200475,  1154736,  4715172,  13856832,   29229255,   43302600,    42869574,    25509168,      6908733;
  7: 1,  84, 2457, 39312,  405405,  2918916, 15324309,  60046272,  177324147,  394053660,   650188539,   773778096,    628694703,    312487308,     71744535;
  8: 1,  96, 3240, 60480,  737100,  6368544, 40864824, 200154240,  759960630, 2251735200,  5201508312,  9285337152,  12573894060,  12499492320,   8609344200,   3673320192,    731794257;
  9: 1, 108, 4131, 88128, 1239300, 12492144, 94732092, 556792704, 2583866142, 9569874600, 28422527562, 67650313536, 128253719412, 191242232496, 219538277100, 187339329792, 111964521321, 41841412812, 7360989291;
%F Formula
The row polynomials satisfy (1+(6*n-3)*x)*P(n,x) = (1+(6*n+9)*x+(36*n+27)*x^2+(54*n+27)*x^3)*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[((1 + (6*n + 9)*x + (36*n + 27)*x^2 + (54*n + 27)*x^3)*P[n - 1, x])/(1 + (6*n - 3)*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] = ((1 + (6*n + 9)*x + (36*n + 27)*x^2 + (54*n + 27)*x^3)*v[i - 1])/(1 + (6*n - 3)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.