A225476 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(2\,x - x^{2} - x^{3}\right)\,P_{n-2}'(x) + \left(4\,x\right)\,P_{n-1}(x) + \left(2\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	16	18						35
3	1	52	162	96					311
4	1	160	1044	1536	600				3341
5	1	484	5940	16320	15000	4320			42065
6	1	1456	31878	145920	237000	155520	35280		607055
7	1	4372	165942	1191456	3045000	3417120	1728720	322560	9875171

All 15 rows

Technical details

Node id: n-39c637c8851b67ce · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-58994edd1d2e3e50

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: 1,     4;
  2: 1,    16,      18;
  3: 1,    52,     162,       96;
  4: 1,   160,    1044,     1536,       600;
  5: 1,   484,    5940,    16320,     15000,      4320;
  6: 1,  1456,   31878,   145920,    237000,    155520,      35280;
  7: 1,  4372,  165942,  1191456,   3045000,   3417120,    1728720,    322560;
  8: 1, 13120,  849384,  9225216,  34851600,  59512320,   50379840,  20643840,   3265920;
  9: 1, 39364, 4305960, 69106560, 371322000, 905567040, 1140955200, 770273280, 264539520, 36288000;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x - 2*x^2)*P[n - 2, x] + (2*x - x^2 - x^3)*D[P[n - 2, x], x] + 4*x*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], 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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x - 2*x^2)*v[i - 2] + (2*x - x^2 - x^3)*deriv(v[i - 2], x) + 4*x*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], 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

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 Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.