A227450 → reverse → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(8\,x - 4\,n - 4\,n^{2} - 7\,n^{2} x^{2} - 8\,n^{3} x - 2\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 10\,x^{2} + 8\,n^{2} x^{2} - 13\,n^{2} x^{3} + 8\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{3} - 3\,n^{2} x^{4} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(2\,n + 4\,n^{2} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	4	8							12
2	8	48	18						74
3	16	192	252	32					492
4	32	640	1800	960	50				3482
5	64	1920	9360	11520	3100	72			26036
6	128	5376	40320	89600	60200	9072	98		204794
7	256	14336	153216	537600	680400	278208	24892	128	1689036

All 15 rows

Technical details

Node id: n-e8998a14bcfa2fbb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e6c66b4f5fbcf5aa

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:    2;
  1:    4,     8;
  2:    8,    48,      18;
  3:   16,   192,     252,       32;
  4:   32,   640,    1800,      960,       50;
  5:   64,  1920,    9360,    11520,     3100,       72;
  6:  128,  5376,   40320,    89600,    60200,     9072,       98;
  7:  256, 14336,  153216,   537600,   680400,   278208,    24892,     128;
  8:  512, 36864,  532224,  2709504,  5560800,  4475520,  1185800,   65280,    162;
  9: 1024, 92160, 1728000, 12042240, 36523200, 48988800, 26738320, 4776960, 165564, 200;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -(4*n^2+4*n+(8*n^3-8)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - (4*x-(8*n^3+8*n^2+10)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) + (2*x^3-(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (4*n^2+2*n+4*n^3*x)*P(n-1,x) + (2*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 + 4*n + (8*n^3 - 8)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - (4*x - (8*n^3 + 8*n^2 + 10)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] + (2*x^3 - (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (4*n^2 + 2*n + 4*n^3*x)*P[n - 1, x] + (2*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 + 4*n + (8*n^3 - 8)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - (4*x - (8*n^3 + 8*n^2 + 10)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) + (2*x^3 - (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (4*n^2 + 2*n + 4*n^3*x)*v[i - 1] + (2*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
    
    ↑ Liu–Wang theorem (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.