Veronese parent of A098172 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 - 5\,x + 2\,n + 3\,n x\right)\,P_{n}(x) = \left(-x^{2} - 3\,x^{3} - 2\,x^{4} + 2\,n x^{2} + 5\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	x²⁷	x²⁸	x²⁹	x³⁰	x³¹	x³²	x³³	x³⁴	x³⁵	x³⁶	x³⁷	x³⁸	x³⁹	x⁴⁰	x⁴¹	x⁴²	P_n(1)
1	1																																											1
2	0	0	3	4																																								7
3	0	0	0	0	5	12	7																																					24
4	0	0	0	0	0	0	7	24	27	10																																		68
5	0	0	0	0	0	0	0	0	9	40	66	48	13																															176
6	0	0	0	0	0	0	0	0	0	0	11	60	130	140	75	16																												432
7	0	0	0	0	0	0	0	0	0	0	0	0	13	84	225	320	255	108	19																									1024
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	15	112	357	630	665	420	147	22																						2368
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	17	144	532	1120	1470	1232	644	192	25																			5376
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	19	180	756	1848	2898	3024	2100	936	243	28																12032
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	21	220	1035	2880	5250	6552	5670	3360	1305	300	31													26624
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	23	264	1375	4290	8910	12936	13398	9900	5115	1760	363	34										58368
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	25	312	1782	6160	14355	23760	28644	25344	16335	7480	2310	432	37							126976
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	27	364	2262	8580	22165	41184	56628	58344	45045	25740	10582	2964	507	40				274432
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	29	420	2821	11648	33033	68068	105105	123552	111111	76076	39039	14560	3731	588	43	589824

Technical details

Node id: n-3d28544a90bbf61e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-1fcba3ddf21b6a9d

Observed on the stored rows ?

Degrees: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 (affine: slope 3, intercept -3)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0, 0, 3, 4;
   3: 0, 0, 0, 0, 5, 12, 7;
   4: 0, 0, 0, 0, 0,  0, 7, 24, 27, 10;
   5: 0, 0, 0, 0, 0,  0, 0,  0,  9, 40, 66, 48,  13;
   6: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 11, 60, 130, 140,  75,  16;
   7: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  13,  84, 225, 320, 255, 108,  19;
   8: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  15, 112, 357, 630, 665,  420,  147,   22;
   9: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  17, 144, 532, 1120, 1470, 1232,  644,  192,   25;
  10: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  19,  180,  756, 1848, 2898, 3024, 2100, 936, 243, 28;

%F Formula

The row polynomials satisfy (2*n-3+(3*n-5)*x)*P(n,x) = ((2*n-1)*x^2+(5*n-3)*x^3+(3*n-2)*x^4)*P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x^2 + (5*n - 3)*x^3 + (3*n - 2)*x^4)*P[n - 1, x])/(2*n - 3 + (3*n - 5)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((2*n - 1)*x^2 + (5*n - 3)*x^3 + (3*n - 2)*x^4)*v[i - 1])/(2*n - 3 + (3*n - 5)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences, Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences, Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences, Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

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.