Real-rooted atlas

A120910 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3 + 9\,x + 9\,x^{2} + 3\,x^{3} + 9\,n x + 15\,n x^{2} + 6\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
193645
227216243486
381972218712964536
424338881312215552607538880
572914580656101166409112526244314928
62187524882952456998408201254723921071632449440
765611837081240029367416057408754960116225042341990418475776

All 15 rows

Technical details
Node id
n-86ae3baaaf435159 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:     3;
  1:     9,      36;
  2:    27,     216,      243;
  3:    81,     972,     2187,     1296;
  4:   243,    3888,    13122,    15552,      6075;
  5:   729,   14580,    65610,   116640,     91125,     26244;
  6:  2187,   52488,   295245,   699840,    820125,    472392,    107163;
  7:  6561,  183708,  1240029,  3674160,   5740875,   4960116,   2250423,    419904;
  8: 19683,  629856,  4960116, 17635968,  34445250,  39680928,  27005076,  10077696,  1594323;
  9: 59049, 2125764, 19131876, 79361856, 186004350, 267846264, 243045684, 136048896, 43046721, 5904900;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (3+(9*n+9)*x+(3*n^2+15*n+9)*x^2+(3*n^2+6*n+3)*x^3)*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[((3 + (9*n + 9)*x + (3*n^2 + 15*n + 9)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
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] = ((3 + (9*n + 9)*x + (3*n^2 + 15*n + 9)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*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.