A225469 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - 12\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-24\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-16\,x^{2} + 16\,x^{3}\right)\,P_{n-2}''(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	4							7
3	9	40	9						58
4	27	316	189	16					548
5	81	2320	2790	576	25				5792
6	243	16564	35910	13600	1375	36			67728
7	729	116920	432279	267840	47375	2808	49		868000
8	2187	821356	5018139	4786096	1318625	132804	5145	64	12084416

All 15 rows

Technical details

Node id: n-26057d10e4649dfa · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e2e41084f3a7d696

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,        4;
   3:     9,       40,         9;
   4:    27,      316,       189,         16;
   5:    81,     2320,      2790,        576,        25;
   6:   243,    16564,     35910,      13600,      1375,        36;
   7:   729,   116920,    432279,     267840,     47375,      2808,       49;
   8:  2187,   821356,   5018139,    4786096,   1318625,    132804,     5145,     64;
   9:  6561,  5758240,  57047580,   80712576,  32532150,   4953312,   319676,   8704,    81;
  10: 19683, 40333924, 640479420, 1312106560, 744224250, 160772472, 15373260, 687360, 13851, 100;

%F Formula

The row polynomials satisfy P(n,x) = (3-12*x+9*x^2)*P(n-2,x) - (24*x-12*x^2-4*x^3)*d/dx P(n-2,x) - (16*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + (8*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(3 - 12*x + 9*x^2)*P[n - 2, x] - (24*x - 12*x^2 - 4*x^3)*D[P[n - 2, x], x] - (16*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + (8*x - 4*x^2)*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 + 4*x; for(n = 3, N, my(i = n); v[i] = (3 - 12*x + 9*x^2)*v[i - 2] - (24*x - 12*x^2 - 4*x^3)*deriv(v[i - 2], x) - (16*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + (8*x - 4*x^2)*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, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - 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.