A137320 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-256 + 256\,x + 144\,x^{2} + 256\,n - 128\,n x - 96\,n^{2} + 16\,n^{2} x + 16\,n^{3} - n^{4}\right)\,P_{n-4}(x) + \left(-256\,x^{2} + 64\,x^{3} + 128\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-4}'(x) + \left(-175 + 112\,x + 148\,n - 32\,n x - 42\,n^{2} + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-112\,x^{2} + 32\,n x^{2}\right)\,P_{n-3}'(x) + \left(-55 + 16\,x + 36\,n - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2}\right)\,P_{n-2}'(x) + \left(-10 + 4\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 16\,x\)

\(P_{3}(x) = 48\,x\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1					0
2	0	16			16
3	0	48			48
4	0	176	144		320
5	0	800	1440		2240
6	0	4384	12240	1024	17648
7	0	28224	105840	21504	155568

All 15 rows

Technical details

Node id: n-e9f23e94d8b75247 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d9d71b71ec22e1b5

Observed on the stored rows ?

Degrees: 0, -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 0;
  2: 0,      16;
  3: 0,      48;
  4: 0,     176,     144;
  5: 0,     800,    1440;
  6: 0,    4384,   12240,    1024;
  7: 0,   28224,  105840,   21504;
  8: 0,  209088,  974736,  329728,   6400;
  9: 0, 1753344, 9688896, 4644864, 230400;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(16*n^2-128*n+256)*x-144*x^2)*P(n-4,x) - ((16*n^2-128*n+256)*x^2-64*x^3)*d/dx P(n-4,x) + (4*n^3-42*n^2+148*n-175-(32*n-112)*x)*P(n-3,x) + (32*n-112)*x^2*d/dx P(n-3,x) - (6*n^2-36*n+55-16*x)*P(n-2,x) - 16*x^2*d/dx P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 0, P(2,x) = 16*x, P(3,x) = 48*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 0;
P[2, x_] := 16*x;
P[3, x_] := 48*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (16*n^2 - 128*n + 256)*x - 144*x^2)*P[n - 4, x] - ((16*n^2 - 128*n + 256)*x^2 - 64*x^3)*D[P[n - 4, x], x] + (4*n^3 - 42*n^2 + 148*n - 175 - (32*n - 112)*x)*P[n - 3, x] + (32*n - 112)*x^2*D[P[n - 3, x], x] - (6*n^2 - 36*n + 55 - 16*x)*P[n - 2, x] - 16*x^2*D[P[n - 2, x], x] + (4*n - 10)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 0; v[3] = 16*x; v[4] = 48*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (16*n^2 - 128*n + 256)*x - 144*x^2)*v[i - 4] - ((16*n^2 - 128*n + 256)*x^2 - 64*x^3)*deriv(v[i - 4], x) + (4*n^3 - 42*n^2 + 148*n - 175 - (32*n - 112)*x)*v[i - 3] + (32*n - 112)*x^2*deriv(v[i - 3], x) - (6*n^2 - 36*n + 55 - 16*x)*v[i - 2] - 16*x^2*deriv(v[i - 2], x) + (4*n - 10)*v[i - 1]); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from veronese_section_2_0 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n …?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[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
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.