Veronese parent of A098172 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 28\,x + 25\,x^{2} - 12\,n - 37\,n x - 30\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(x^{2} + 4\,x^{3} + 7\,x^{4} + 4\,x^{5} - 4\,n x^{2} - 17\,n x^{3} - 25\,n x^{4} - 12\,n x^{5} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 21\,n^{2} x^{4} + 9\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

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

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	9	16																																								25
3	0	0	0	0	25	72	49																																					146
4	0	0	0	0	0	0	49	192	243	100																																		584
5	0	0	0	0	0	0	0	0	81	400	726	576	169																															1952
6	0	0	0	0	0	0	0	0	0	0	121	720	1690	1960	1125	256																												5872
7	0	0	0	0	0	0	0	0	0	0	0	0	169	1176	3375	5120	4335	1944	361																									16480
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	225	1792	6069	11340	12635	8400	3087	484																						44032
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	289	2592	10108	22400	30870	27104	14812	4608	625																			113408
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	361	3600	15876	40656	66654	72576	52500	24336	6561	784																283904
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	441	4840	23805	69120	131250	170352	153090	94080	37845	9000	961													694784
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	529	6336	34375	111540	240570	362208	388542	297000	158565	56320	11979	1156										1669120
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	625	8112	48114	172480	416295	712800	887964	811008	539055	254320	80850	15552	1369							3948544
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	729	10192	65598	257400	687115	1317888	1868724	1983696	1576575	926640	391534	112632	19773	1600				9220096
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	841	12600	87451	372736	1090089	2314312	3678675	4447872	4111107	2890888	1522521	582400	152971	24696	1849	21291008

Technical details

Node id: n-9e444e7404c8ff14 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7d7bdc1257ffa7e8

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 0, 0, 9, 16;
   3: 0, 0, 0,  0, 25, 72, 49;
   4: 0, 0, 0,  0,  0,  0, 49, 192, 243, 100;
   5: 0, 0, 0,  0,  0,  0,  0,   0,  81, 400, 726, 576,  169;
   6: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0, 121, 720, 1690, 1960, 1125,  256;
   7: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,  169, 1176, 3375, 5120, 4335,  1944,   361;
   8: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,  225, 1792, 6069, 11340, 12635,  8400,  3087,   484;
   9: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,  289,  2592, 10108, 22400, 30870, 27104, 14812,  4608,   625;
  10: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,    0,     0,   361,  3600, 15876, 40656, 66654, 72576, 52500, 24336, 6561, 784;

%F Formula

The row polynomials satisfy (4*n^2-12*n+9+(12*n^2-37*n+28)*x+(9*n^2-30*n+25)*x^2)*P(n,x) = ((4*n^2-4*n+1)*x^2+(16*n^2-17*n+4)*x^3+(21*n^2-25*n+7)*x^4+(9*n^2-12*n+4)*x^5)*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[(((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*P[n - 1, x])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*x^2)]];
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] = (((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*v[i - 1])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*x^2)); 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
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
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

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

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.