A269945 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	1	3	1						5
4	1	11	8	1					21
5	1	43	63	18	1				126
6	1	171	526	267	35	1			1001
7	1	683	4565	3916	902	61	1		10129
8	1	2731	40404	58773	21450	2548	98	1	126006

All 15 rows

Technical details

Node id: n-395052ed1a3a82d1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a5a3042851370ae6

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: 1,     1;
   3: 1,     3,       1;
   4: 1,    11,       8,        1;
   5: 1,    43,      63,       18,        1;
   6: 1,   171,     526,      267,       35,       1;
   7: 1,   683,    4565,     3916,      902,      61,      1;
   8: 1,  2731,   40404,    58773,    21450,    2548,     98,     1;
   9: 1, 10923,  360907,   902694,   503503,   93158,   6258,   148,   1;
  10: 1, 43691, 3237242, 14093119, 11921845, 3242291, 336464, 13770, 213, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (x^2 - x^3)*v[i - 2] - x^3*deriv(v[i - 2], x) - x^4*deriv(deriv(v[i - 2], x), x) + (1 - x + x^2)*v[i - 1] + 3*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) =…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.