A272866 → D → D

nonnegative coefficients

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

Recurrence

\(\left(16 + 18\,x + 6\,x^{2} - 25\,n - 30\,n x - 10\,n x^{2} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-7\,n - 27\,n x - 27\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} + 9\,n^{2} + 39\,n^{2} x + 49\,n^{2} x^{2} + 24\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	P_n(1)
0																												0
1	2																											2
2	22	36	12																									70
3	60	270	360	180	30																							900
4	116	864	2340	2880	1740	504	56																					8500
5	190	1980	8220	17460	20550	13860	5320	1080	90																			68750
6	282	3780	21240	65160	119670	136836	99120	45360	12690	1980	132																	506250
7	392	6426	45612	183960	465990	776286	869848	662256	342090	117810	25872	3276	182															3500000
8	520	10080	86520	433440	1407840	3119760	4844840	5348160	4223520	2383920	951720	262080	47320	5040	240													23125000
9	666	14904	150120	899640	3574620	9942912	19951848	29383560	32065470	26040960	15728328	7017192	2276820	521640	79920	7344	306											147656250
10	830	21060	243540	1699920	7998300	26845560	66396120	123385680	174319290	188505900	156505140	99712080	48523020	17849160	4870800	954720	126990	10260	380									917968750
11	1012	28710	374880	2987820	16256790	64008252	188708520	425505960	743723640	1016182530	1090794672	921929580	613302690	320041260	130054320	40634352	9559440	1636470	192280	13860	462							5585937500
12	1212	38016	553212	4957920	30639180	138561192	475087536	1263110112	2642652540	4393247760	5839846452	6230424096	5344030692	3684071160	2036089440	897348672	312519636	84780432	17518380	2661120	279972	18216	552					33398437500
13	1430	49140	788580	7850700	54332850	277657380	1086525440	3333124080	8142849000	16012725300	25531428780	33165758340	35202424530	30569748300	21714264000	12591802080	5937085440	2260924380	688216100	164864700	30360330	4144140	394680	23400	650			196777343750
14	1666	62244	1092000	11957400	91632450	522341820	2298576280	7999992000	22388265900	50962111200	95120085060	146395234440	186485003250	197070507900	172945609200	126015402240	76120104060	37999962000	15597481900	5223418200	1411139730	302522220	50232000	6224400	541450	29484	756	1145019531250

Technical details

Node id: n-2303cf9bfb74fbab · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6c1b3ac9ecf30630

Observed on the stored rows ?

Degrees: -1, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 (affine: slope 2, intercept -2)
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:   2;
  2:  22,    36,     12;
  3:  60,   270,    360,    180,      30;
  4: 116,   864,   2340,   2880,    1740,     504,       56;
  5: 190,  1980,   8220,  17460,   20550,   13860,     5320,     1080,       90;
  6: 282,  3780,  21240,  65160,  119670,  136836,    99120,    45360,    12690,     1980,      132;
  7: 392,  6426,  45612, 183960,  465990,  776286,   869848,   662256,   342090,   117810,    25872,    3276,     182;
  8: 520, 10080,  86520, 433440, 1407840, 3119760,  4844840,  5348160,  4223520,  2383920,   951720,  262080,   47320,   5040,   240;
  9: 666, 14904, 150120, 899640, 3574620, 9942912, 19951848, 29383560, 32065470, 26040960, 15728328, 7017192, 2276820, 521640, 79920, 7344, 306;

%F Formula

The row polynomials satisfy (9*n^2-25*n+16+(12*n^2-30*n+18)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (9*n^2-7*n+(39*n^2-27*n)*x+(49*n^2-27*n)*x^2+(24*n^2-12*n)*x^3+(4*n^2-2*n)*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 7*n + (39*n^2 - 27*n)*x + (49*n^2 - 27*n)*x^2 + (24*n^2 - 12*n)*x^3 + (4*n^2 - 2*n)*x^4)*P[n - 1, x])/(9*n^2 - 25*n + 16 + (12*n^2 - 30*n + 18)*x + (4*n^2 - 10*n + 6)*x^2)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 7*n + (39*n^2 - 27*n)*x + (49*n^2 - 27*n)*x^2 + (24*n^2 - 12*n)*x^3 + (4*n^2 - 2*n)*x^4)*v[i - 1])/(9*n^2 - 25*n + 16 + (12*n^2 - 30*n + 18)*x + (4*n^2 - 10*n + 6)*x^2)); 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 derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.