Real-rooted atlas

A178820 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 6\,n x + 18\,n^{2} x + 9\,n^{3} x^{2}\right)\,P_{n}(x) = \left(24 + 108\,x + 162\,x^{2} + 81\,x^{3} + 8\,n + 144\,n x + 324\,n x^{2} + 189\,n x^{3} + 36\,n^{2} x + 144\,n^{2} x^{2} + 135\,n^{2} x^{3} + 18\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
184856
2404808101330
316028809720864021400
4560134406804012096070875273875
517925376036288096768011340004898883010000
6537619353616329605806080102060008817984300056429662500
71536064512065318402903040068040000881798406001128016796160269250000

All 15 rows

Technical details
Node id
n-8fc19771ea009e4e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      8,      48;
  2:     40,     480,      810;
  3:    160,    2880,     9720,      8640;
  4:    560,   13440,    68040,    120960,      70875;
  5:   1792,   53760,   362880,    967680,    1134000,     489888;
  6:   5376,  193536,  1632960,   5806080,   10206000,    8817984,    3000564;
  7:  15360,  645120,  6531840,  29030400,   68040000,   88179840,   60011280,   16796160;
  8:  42240, 2027520, 23950080, 127733760,  374220000,  646652160,  660124080,  369515520,   87687765;
  9: 112640, 6082560, 82114560, 510935040, 1796256000, 3879912960, 5280992640, 4434186240, 2104506360, 433026000;
%F Formula
The row polynomials satisfy (4*n+(18*n^2-6*n)*x+9*n^3*x^2)*P(n,x) = (8*n+24+(36*n^2+144*n+108)*x+(18*n^3+144*n^2+324*n+162)*x^2+(27*n^3+135*n^2+189*n+81)*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[((8*n + 24 + (36*n^2 + 144*n + 108)*x + (18*n^3 + 144*n^2 + 324*n + 162)*x^2 + (27*n^3 + 135*n^2 + 189*n + 81)*x^3)*P[n - 1, x])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*x^2)]];
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] = ((8*n + 24 + (36*n^2 + 144*n + 108)*x + (18*n^3 + 144*n^2 + 324*n + 162)*x^2 + (27*n^3 + 135*n^2 + 189*n + 81)*x^3)*v[i - 1])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*x^2)); 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.