Real-rooted atlas

A120910 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 2\,n x\right)\,P_{n}(x) = \left(9 + 12\,x + 4\,x^{2} + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
191221
2277236135
38132432496825
42431296194411522404875
5729486097208640360057628125
6218717496437405184032400103681344159375
7656161236183708272160226800108864282243072890625

All 15 rows

Technical details
Node id
n-4ea1149559d407b2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-925da9e442f58149

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:     3;
  1:     9,     12;
  2:    27,     72,      36;
  3:    81,    324,     324,      96;
  4:   243,   1296,    1944,    1152,     240;
  5:   729,   4860,    9720,    8640,    3600,     576;
  6:  2187,  17496,   43740,   51840,   32400,   10368,    1344;
  7:  6561,  61236,  183708,  272160,  226800,  108864,   28224,   3072;
  8: 19683, 209952,  734832, 1306368, 1360800,  870912,  338688,  73728,   6912;
  9: 59049, 708588, 2834352, 5878656, 7348320, 5878656, 3048192, 995328, 186624, 15360;
%F Formula
The row polynomials satisfy (3+2*n*x)*P(n,x) = (9+(6*n+12)*x+(4*n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (6*n + 12)*x + (4*n + 4)*x^2)*P[n - 1, x])/(3 + 2*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (6*n + 12)*x + (4*n + 4)*x^2)*v[i - 1])/(3 + 2*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. [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.