basis-map coefficients of A371080, shifted → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 30\,x - 9\,x^{2} + 30\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	15	18													33
3	117	216	81												414
4	1254	2862	1782	324											6222
5	17142	44730	36045	11340	1215										110472
6	285192	818532	778005	325620	61965	4374									2273688
7	5593368	17263260	18465489	9298800	2398410	306180	15309								53340816
8	126373776	413351928	483925428	278435556	87635520	15370236	1408428	52488							1406553360
9	3232377360	11092040856	13958219376	8896590612	3235021335	699743772	89006526	6141096	177147						41209318080
10	92332390080	329971408128	440744326380	304937414640	123942812175	31238902422	4941285930	477115920	25686315	590490					1328611932480
11	2913427164480	10783107992448	15147945454356	11216037159360	4985742482325	1414600098912	262516022307	31732145280	2406541995	103926240	1948617				46757620936320
12	100641338824320	384106234730112	563554131414120	441937045235664	211575383746650	66045076299378	13876645104630	1978947873144	188918418150	11555298810	409209570	6377292			1783915692531840
13	3777613254387840	14815778717960064	22579980918607944	18605887199376048	9485553118259790	3205339204564926	744593635919133	120798140236488	13668930543411	1057274116470	53273240163	1575191124	20726199		73350325263129600
14	153098227737369600	615296828244729600	969870240975138048	834555411649473696	449194201728051780	162361559408332284	41002432652742561	7384818075467976	954450945034587	87854066470170	5619934190871	237356430948	5948419113	66961566	3233811888788812800

Technical details

Node id: n-3303bba34ca074bd · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b08aa9c7820883e4

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:          3;
  2:         15,          18;
  3:        117,         216,          81;
  4:       1254,        2862,        1782,        324;
  5:      17142,       44730,       36045,      11340,       1215;
  6:     285192,      818532,      778005,     325620,      61965,      4374;
  7:    5593368,    17263260,    18465489,    9298800,    2398410,    306180,    15309;
  8:  126373776,   413351928,   483925428,  278435556,   87635520,  15370236,  1408428,   52488;
  9: 3232377360, 11092040856, 13958219376, 8896590612, 3235021335, 699743772, 89006526, 6141096, 177147;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*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_A013610 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.