Real-rooted atlas

basis-map coefficients of A026729 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2167
31122740
411881108208
51241624324051024
61302701080202514584864
7136405216060758748510322528
8142567378014175306183572117496102400
91487566048283508164814288413996859049458752
101549729072510301837084286526298565314411968302031616
111601215129608505036741610716302099520265720519683006495398912896
12166148517820133650673596235758657736809743085108256507144929212576438797312
131721782237602004751154736471517213856832292292554330260042869574255091686908733167772160
141782106308882895751876446875674830023136759960631407334501857681541658095928981352922320522721420288
1518424573931240540529189161532430960046272177324147394053660650188539773778096628694703312487308717445353087007744
Technical details
Node id
n-4b82b7f00d162b38 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3151c955f16eaf46

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  6;
   3: 1, 12,  27;
   4: 1, 18,  81,  108;
   5: 1, 24, 162,  432,   405;
   6: 1, 30, 270, 1080,  2025,   1458;
   7: 1, 36, 405, 2160,  6075,   8748,   5103;
   8: 1, 42, 567, 3780, 14175,  30618,  35721,  17496;
   9: 1, 48, 756, 6048, 28350,  81648, 142884, 139968,  59049;
  10: 1, 54, 972, 9072, 51030, 183708, 428652, 629856, 531441, 196830;
%F Formula
The row polynomials satisfy (1+(3*n-3)*x)*P(n,x) = (1+(3*n+3)*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 3)*x + 9*n*x^2)*P[n - 1, x])/(1 + (3*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + (3*n + 3)*x + 9*n*x^2)*v[i - 1])/(1 + (3*n - 3)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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