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

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	20	32													52
3	156	384	192												732
4	1672	5088	4224	1024											12008
5	22856	79520	85440	35840	5120										228776
6	380256	1455168	1844160	1029120	261120	24576									4994400
7	7457824	30690240	43770048	29388800	10106880	1720320	114688								123248800
8	168498368	734847872	1147082496	879993856	369295360	86360064	10551296	524288							3397153600
9	4309836480	19719183744	33086149632	28117619712	13632353280	3931619328	666796032	61341696	2359296						103527259200
10	123109853440	586615836672	1044727292160	963752816640	522293990400	175520636928	37017845760	4765777920	342097920	10485760					3458156633600
11	3884569552640	19169969764352	35906241076992	35448216207360	21009877785600	7948150898688	1966649524224	316963553280	32051036160	1845493760	46137344				125684581030400
12	134188451765760	682855528409088	1335832015203840	1396739303460864	891576925747200	371084543926272	103957454684160	19767158636544	2516071219200	205195837440	9688842240	201326592			4938732539059200
13	5036817672517120	26339162165262336	53522917732996608	58803791642472448	39972042769950720	18009697368858624	5578153694969856	1206618948108288	182046848974848	18774784737280	1261348847616	49727668224	872415232		208671335577779200
14	204130970316492800	1093861027990630400	2298951682311438336	2637607226941546496	1892900669010391040	912253700050108416	307171402186801152	73764889101336576	12711659225481216	1560088496046080	133063006748672	7493174427648	250383171584	3758096384	9435054125952716800

Technical details

Node id: n-472235c6f6d3b99c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f8e7492a8b48ef64

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:          0;
  1:          4;
  2:         20,          32;
  3:        156,         384,         192;
  4:       1672,        5088,        4224,        1024;
  5:      22856,       79520,       85440,       35840,        5120;
  6:     380256,     1455168,     1844160,     1029120,      261120,      24576;
  7:    7457824,    30690240,    43770048,    29388800,    10106880,    1720320,    114688;
  8:  168498368,   734847872,  1147082496,   879993856,   369295360,   86360064,  10551296,   524288;
  9: 4309836480, 19719183744, 33086149632, 28117619712, 13632353280, 3931619328, 666796032, 61341696, 2359296;

%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) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
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] = 0; v[2] = 4; 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.