A225476 → D+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	5	8	2						15
3	14	49	36	6					105
4	41	272	404	192	24				933
5	122	1441	3720	3420	1200	120			10023
6	365	7448	30902	47040	31080	8640	720		126195
7	1094	37969	241836	561666	596400	306600	70560	5040	1821165

All 15 rows

Technical details

Node id: n-362d698441b465c3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0d549fd1b86bbcc3

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:    2,      1;
  2:    5,      8,        2;
  3:   14,     49,       36,        6;
  4:   41,    272,      404,      192,        24;
  5:  122,   1441,     3720,     3420,      1200,       120;
  6:  365,   7448,    30902,    47040,     31080,      8640,       720;
  7: 1094,  37969,   241836,   561666,    596400,    306600,     70560,     5040;
  8: 3281, 192032,  1824104,  6152832,   9659664,   7822080,   3286080,   645120,   40320;
  9: 9842, 966721, 13435920, 63730680, 140626080, 164863440, 107533440, 38162880, 6531840, 362880;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(3 + 2*x - x^2 - x^3)*P[n - 2, x] - (2*x - x^2 - 3*x^3 - x^4)*D[P[n - 2, x], x] + (4 + 2*x - x^2)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 2*x - x^2 - x^3)*v[i - 2] - (2*x - x^2 - 3*x^3 - x^4)*deriv(v[i - 2], x) + (4 + 2*x - x^2)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.