A225473 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 16\,x - 32\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 12\,x^{2} - 16\,x^{3}\right)\,P_{n-2}'(x) + \left(2 + 16\,x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	16							19
3	9	160	288						457
4	27	1264	6048	6144					13483
5	81	9280	89280	221184	153600				473425
6	243	66256	1149120	5222400	8448000	4423680			19309699
7	729	467680	13832928	102850560	291072000	345047040	144506880		897777817
8	2187	3285424	160580448	1837860864	8101632000	16318955520	15173222400	5284823040	46880361883

All 15 rows

Technical details

Node id: n-87220c9644b39788 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-661188e67d3889fb

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,        16;
   3:     9,       160,         288;
   4:    27,      1264,        6048,         6144;
   5:    81,      9280,       89280,       221184,        153600;
   6:   243,     66256,     1149120,      5222400,       8448000,        4423680;
   7:   729,    467680,    13832928,    102850560,     291072000,      345047040,      144506880;
   8:  2187,   3285424,   160580448,   1837860864,    8101632000,    16318955520,    15173222400,     5284823040;
   9:  6561,  23032960,  1825522560,  30993629184,  199877529600,   608662978560,   942762885120,   718735933440,   214035333120;
  10: 19683, 161335696, 20495341440, 503848919040, 4572513792000, 19755721359360, 45337588531200, 56758999449600, 36600041963520, 9512681472000;

%F Formula

The row polynomials satisfy P(n,x) = (3+16*x-32*x^2)*P(n-2,x) + (4*x-12*x^2-16*x^3)*d/dx P(n-2,x) + (2+16*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 16*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[(3 + 16*x - 32*x^2)*P[n - 2, x] + (4*x - 12*x^2 - 16*x^3)*D[P[n - 2, x], x] + (2 + 16*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 16*x; for(n = 3, N, my(i = n); v[i] = (3 + 16*x - 32*x^2)*v[i - 2] + (4*x - 12*x^2 - 16*x^3)*deriv(v[i - 2], x) + (2 + 16*x)*v[i - 1] + (4*x + 4*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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!*S_4(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.