A225467 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 8\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 16\,x^{2}\right)\,P_{n-2}'(x) + \left(2 + 8\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	8							11
3	9	80	48						137
4	27	632	1008	256					1923
5	81	4640	14880	9216	1280				30097
6	243	33128	191520	217600	70400	6144			519035
7	729	233840	2305488	4285440	2425600	479232	28672		9759001
8	2187	1642712	26763408	76577536	67513600	22665216	3010560	131072	198306291

All 15 rows

Technical details

Node id: n-57f5c09ed79c8f49 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9135fad28cbf1932

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:     1;
   2:     3,        8;
   3:     9,       80,         48;
   4:    27,      632,       1008,         256;
   5:    81,     4640,      14880,        9216,        1280;
   6:   243,    33128,     191520,      217600,       70400,        6144;
   7:   729,   233840,    2305488,     4285440,     2425600,      479232,      28672;
   8:  2187,  1642712,   26763408,    76577536,    67513600,    22665216,    3010560,     131072;
   9:  6561, 11516480,  304253760,  1291401216,  1665646080,   845365248,  187056128,   17825792,    589824;
  10: 19683, 80667848, 3415890240, 20993704960, 38104281600, 27438501888, 8995553280, 1407713280, 100859904, 2621440;

%F Formula

The row polynomials satisfy P(n,x) = (3+8*x-16*x^2)*P(n-2,x) + (4*x-16*x^2)*d/dx P(n-2,x) + (2+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 8*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[(3 + 8*x - 16*x^2)*P[n - 2, x] + (4*x - 16*x^2)*D[P[n - 2, x], x] + (2 + 8*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 8*x; for(n = 3, N, my(i = n); v[i] = (3 + 8*x - 16*x^2)*v[i - 2] + (4*x - 16*x^2)*deriv(v[i - 2], x) + (2 + 8*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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
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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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 Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.