A166960 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	6	1						8
3	1	21	27	1					50
4	1	58	270	112	1				442
5	1	141	1738	2878	453	1			5212
6	1	318	8739	39320	28167	1818	1		78364
7	1	685	37665	375755	769955	264411	7279	1	1455752

All 15 rows

Technical details

Node id: n-0b73360ed752ea1a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2b1205fb0a5fe514

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,    1;
  2: 1,    6,      1;
  3: 1,   21,     27,        1;
  4: 1,   58,    270,      112,         1;
  5: 1,  141,   1738,     2878,       453,         1;
  6: 1,  318,   8739,    39320,     28167,      1818,         1;
  7: 1,  685,  37665,   375755,    769955,    264411,      7279,        1;
  8: 1, 1434, 146560,  2858960,  13243650,  13905746,   2430652,    29124,      1;
  9: 1, 2949, 531456, 18617280, 169140810, 414525726, 239506500, 22108860, 116505, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n^2*x)*P[n - 1, x] + (x - (2*n - 1)*x^2)*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n^2*x)*v[i - 1] + (x - (2*n - 1)*x^2)*deriv(v[i - 1], x) + x^3*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 - 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A166960: Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.

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.