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

\(\left(1 + 4\,x + 9\,x^{2} - 4\,n - 17\,n x - 18\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(x^{2} - x^{4} + 4\,n x^{2} + 11\,n x^{3} + 7\,n x^{4} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 21\,n^{2} x^{4} + 9\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 9\,x^{2}\)

Rows ?

n	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	x⁴³	x⁴⁴	P_n(1)
1	9																																											9
2	0	0	25	36																																								61
3	0	0	0	0	49	128	81																																					258
4	0	0	0	0	0	0	81	300	363	144																																		888
5	0	0	0	0	0	0	0	0	121	576	1014	784	225																															2720
6	0	0	0	0	0	0	0	0	0	0	169	980	2250	2560	1445	324																												7728
7	0	0	0	0	0	0	0	0	0	0	0	0	225	1536	4335	6480	5415	2400	441																									20832
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	289	2268	7581	14000	15435	10164	3703	576																						54016
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	361	3200	12348	27104	37030	32256	17500	5408	729																			135936
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	441	4356	19044	48384	78750	85176	61236	28224	7569	900																334080
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	529	5760	28125	81120	153090	197568	176610	108000	43245	10240	1089													805376
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	625	7436	40095	129360	277530	415800	443982	337920	179685	63580	13475	1296										1910784
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	729	9408	55506	198000	475695	811008	1006236	915552	606375	285120	90354	17328	1521							4472832
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	841	11700	74958	292864	778635	1487772	2102100	2223936	1761903	1032460	435006	124800	21853	1764				10350592
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	961	14336	99099	420784	1226225	2594592	4111107	4955808	4567563	3203200	1682681	642096	168259	27104	2025	23715840

Technical details

Node id: n-23dd1c780661a15d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d772db5482e65522

Observed on the stored rows ?

Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0, 0, 9;
   2: 0, 0, 0, 0, 25, 36;
   3: 0, 0, 0, 0,  0,  0, 49, 128, 81;
   4: 0, 0, 0, 0,  0,  0,  0,   0, 81, 300, 363, 144;
   5: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0, 121, 576, 1014, 784,  225;
   6: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,  169, 980, 2250, 2560, 1445,  324;
   7: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,  225, 1536, 4335, 6480, 5415,  2400,   441;
   8: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,  289, 2268, 7581, 14000, 15435, 10164,  3703,   576;
   9: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,  361,  3200, 12348, 27104, 37030, 32256, 17500,  5408,   729;
  10: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,    0,     0,   441,  4356, 19044, 48384, 78750, 85176, 61236, 28224, 7569, 900;

%F Formula

The row polynomials satisfy (4*n^2-4*n+1+(12*n^2-17*n+4)*x+(9*n^2-18*n+9)*x^2)*P(n,x) = ((4*n^2+4*n+1)*x^2+(16*n^2+11*n)*x^3+(21*n^2+7*n-1)*x^4+9*n^2*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 9*x^2.

%t Mathematica

P[1, x_] := 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*P[n - 1, x])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 9*x^2; for(n = 2, N, my(i = n); v[i] = (((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*v[i - 1])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*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).?
  - 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.