basis-map coefficients of A371080, shifted → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 12\,x - x^{2} + 36\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	3	5	1													9
3	18	36	12	1												67
4	162	360	153	22	1											698
5	1944	4644	2295	435	35	1										9354
6	29160	73548	40419	9135	990	51	1									153304
7	524880	1382184	823284	210924	27720	1953	70	1								2971016
8	11022480	30075624	19099800	5384232	812889	70308	3486	92	1							66468912
9	264539520	743859936	497997396	151512444	25441857	2553201	156870	5778	117	1						1686067120
10	7142567040	20613297312	14429449836	4674945780	854792595	96196653	6911163	317790	9045	145	1					47818487360
11	214277011200	632684053440	460232157528	157242518136	30846286350	3806138160	308656656	16700013	596970	13530	176	1				1499414132160
12	7071141369600	21307127785920	16029548772984	5732911468152	1193561544186	158939787600	14211007173	872536698	36890073	1055340	19503	210	1			51508352237440
13	254561089305600	781198883032320	605488876821504	225384880148064	49391725295484	7015986559296	680260566843	46263254895	2229530589	75764403	1774773	27261	247	1		1923770272081280
14	9927882482918400	30975878616871680	24658583975934336	9509555862582048	2179494221712444	327327135159024	33996824084223	2517325386147	134898119928	5245048809	146486340	2862405	37128	287	1	77615376737203200

Technical details

Node id: n-9329f5c4405ead44 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1d885b2b997ee023

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:         3,         5,         1;
  3:        18,        36,        12,         1;
  4:       162,       360,       153,        22,        1;
  5:      1944,      4644,      2295,       435,       35,       1;
  6:     29160,     73548,     40419,      9135,      990,      51,      1;
  7:    524880,   1382184,    823284,    210924,    27720,    1953,     70,    1;
  8:  11022480,  30075624,  19099800,   5384232,   812889,   70308,   3486,   92,   1;
  9: 264539520, 743859936, 497997396, 151512444, 25441857, 2553201, 156870, 5778, 117, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-36*n+36+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, 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 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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 basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
  - Parent-row provers of polytool polytool ? step › direct check

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.