basis-map coefficients of A371080, shifted → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 40\,x - 16\,x^{2} + 30\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 8\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	4							9
2	24	52	16						92
3	184	540	384	64					1172
4	1952	6760	6768	2432	256				18168
5	26496	102376	125200	64320	14080	1024			333496
6	438496	1835424	2571744	1643840	518400	76800	4096		7088800
7	8564384	38148064	59115168	43978816	17454080	3741696	401408	16384	171420000

All 15 rows

Technical details

Node id: n-d7007867f0a138dc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ef507ae4352d333a

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:          5,           4;
  2:         24,          52,          16;
  3:        184,         540,         384,          64;
  4:       1952,        6760,        6768,        2432,         256;
  5:      26496,      102376,      125200,       64320,       14080,       1024;
  6:     438496,     1835424,     2571744,     1643840,      518400,      76800,       4096;
  7:    8564384,    38148064,    59115168,    43978816,    17454080,    3741696,     401408,     16384;
  8:  192842688,   903346240,  1514506432,  1262354688,   589293824,  160219136,   24944640,   2031616,    65536;
  9: 4918444480, 24029020224, 42945741504, 39146336256, 20661758208, 6658089984, 1322065920, 156598272, 10027008, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(24*n-40)*x+16*x^2)*P(n-2,x) + (6*n-7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*P[n - 2, x] + (6*n - 7 + 8*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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*v[i - 2] + (6*n - 7 + 8*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_A013611 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.