A299989 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4	3													8
2	1	8	22	24	9											64
3	1	12	57	136	171	108	27									512
4	1	16	108	400	886	1200	972	432	81							4096
5	1	20	175	880	2810	5944	8430	7920	4725	1620	243					32768
6	1	24	258	1640	6855	19824	40636	59472	61695	44280	20898	5832	729			262144
7	1	28	357	2744	14189	52164	140497	281488	421491	469476	383103	222264	86751	20412	2187	2097152

All 15 rows

Technical details

Node id: n-45c3e3fbc36932e1 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-45b1586921223be0

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  4,   3;
  2: 1,  8,  22,   24,     9;
  3: 1, 12,  57,  136,   171,    108,     27;
  4: 1, 16, 108,  400,   886,   1200,    972,     432,      81;
  5: 1, 20, 175,  880,  2810,   5944,   8430,    7920,    4725,     1620,      243;
  6: 1, 24, 258, 1640,  6855,  19824,  40636,   59472,   61695,    44280,    20898,     5832,      729;
  7: 1, 28, 357, 2744, 14189,  52164, 140497,  281488,  421491,   469476,   383103,   222264,    86751,    20412,     2187;
  8: 1, 32, 472, 4256, 26236, 117152, 391720,  999968, 1968934,  2999904,  3525480,  3163104,  2125116,  1034208,   344088,   69984,    6561;
  9: 1, 36, 603, 6240, 44676, 234864, 939036, 2918304, 7143966, 13875544, 21431898, 26264736, 25353972, 19023984, 10856268, 4548960, 1318761, 236196, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x + 3*x^2)*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - 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

Related ?

Same fingerprint ?

A299989: Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(…

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.