basis-map coefficients of A371080, shifted → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 30\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	14	18	4													36
3	106	174	72	8												360
4	1116	2108	1164	240	16											4644
5	15068	31308	20620	5800	720	32										73548
6	248368	553920	412416	141160	24240	2016	64									1382184
7	4835472	11401472	9307536	3660552	775040	90048	5376	128								30075624
8	108593504	267961152	235241296	102794720	25290384	3657472	307328	13824	256							743859936
9	2763526240	7084714176	6600666672	3135940800	865348848	146633760	15447936	983808	34560	512						20613297312
10	78595950720	208208885888	203917891104	103764854880	31380324960	6006847392	746286912	59930880	2995200	84480	1024					632684053440
11	2470556520320	6734777217408	6884526355168	3711606891168	1210437059040	255501593952	36062759040	3433060224	217367040	8757760	202752	2048				21307127785920
12	85055465178880	237808107985152	252335942951616	142955873766464	49685830546176	11370551784768	1778533614528	193239408384	14566134528	746408960	24769536	479232	4096			781198883032320
13	3182974690210560	9103199377574144	9982759987282368	5905358549862720	2167935580974592	531226788700608	90479579306688	10938973229184	944729660160	57841445376	2449368064	68130816	1118208	8192		30975878616871680
14	128648119316326400	375530742155904000	424101588787758592	260640105113489920	100365758827536384	26084593244597888	4775437256798208	630969323707008	60845471681280	4282130524672	217324563456	7735949312	183013376	2580480	16384	1320842666874447360

Technical details

Node id: n-5d19d732b0c7e06c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8032bdd3ad4ca3e6

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:          3,          2;
  2:         14,         18,          4;
  3:        106,        174,         72,          8;
  4:       1116,       2108,       1164,        240,        16;
  5:      15068,      31308,      20620,       5800,       720,        32;
  6:     248368,     553920,     412416,     141160,     24240,      2016,       64;
  7:    4835472,   11401472,    9307536,    3660552,    775040,     90048,     5376,    128;
  8:  108593504,  267961152,  235241296,  102794720,  25290384,   3657472,   307328,  13824,   256;
  9: 2763526240, 7084714176, 6600666672, 3135940800, 865348848, 146633760, 15447936, 983808, 34560, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(12*n-20)*x+4*x^2)*P(n-2,x) + (6*n-7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*P[n - 2, x] + (6*n - 7 + 4*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*v[i - 2] + (6*n - 7 + 4*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) 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
  - Geometric basis map (elementary) 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 (above)

References

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