A269945 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(x + 5\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(5\,x^{2} + 9\,x^{3}\right)\,P_{n-2}''(x) + \left(2\,x^{3} + 4\,x^{4}\right)\,P_{n-2}'''(x) + \left(1 + 4\,x\right)\,P_{n-1}(x) + \left(-x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	4														4
3	0	4	9													13
4	0	4	45	16												65
5	0	4	189	224	25											442
6	0	4	765	2352	750	36										3907
7	0	4	3069	22528	15675	1980	49									43305
8	0	4	12285	208208	286000	72072	4459	64								583092
9	0	4	49149	1895712	4901325	2213640	258622	8960	81							9327493
10	0	4	196605	17148784	81383250	62398908	12323402	776832	16524	100						174244409
11	0	4	786429	154688576	1328926975	1677164580	528574319	54160640	2040714	28500	121					3746370858
12	0	4	3145725	1393595280	21504532500	43842769344	21311482829	3344254144	199152756	4827900	46585	144				91603807211
13	0	4	12582909	12547949920	346250012625	1127035760400	826888262340	191703859200	16978348035	636927500	10500259	72864	169			2522064276225
14	0	4	50331645	112953918896	5559606373750	28674494028180	31301998340340	10473506423040	1329239471040	72552051000	1820708175	21312720	109850	196		77526243068836
15	0	4	201326589	1016674748544	89130192478275	724868183882700	1165901112679485	554086057459200	98326857713220	7517752515000	269858799210	4745632320	40831245	160524	225	2641121678226541

Technical details

Node id: n-8f9c04795daca07f · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-2d3678d272a636c9

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: 0, 4;
   3: 0, 4,      9;
   4: 0, 4,     45,       16;
   5: 0, 4,    189,      224,       25;
   6: 0, 4,    765,     2352,      750,       36;
   7: 0, 4,   3069,    22528,    15675,     1980,       49;
   8: 0, 4,  12285,   208208,   286000,    72072,     4459,     64;
   9: 0, 4,  49149,  1895712,  4901325,  2213640,   258622,   8960,    81;
  10: 0, 4, 196605, 17148784, 81383250, 62398908, 12323402, 776832, 16524, 100;

%F Formula

The row polynomials satisfy P(n,x) = (4*x+9*x^2)*P(n-2,x) + (x+5*x^2+4*x^3)*d/dx P(n-2,x) + (5*x^2+9*x^3)*d^2/dx^2 P(n-2,x) + (2*x^3+4*x^4)*d^3/dx^3 P(n-2,x) + (1+4*x)*P(n-1,x) - (x+4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[(4*x + 9*x^2)*P[n - 2, x] + (x + 5*x^2 + 4*x^3)*D[P[n - 2, x], x] + (5*x^2 + 9*x^3)*D[P[n - 2, x], {x, 2}] + (2*x^3 + 4*x^4)*D[P[n - 2, x], {x, 3}] + (1 + 4*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = (4*x + 9*x^2)*v[i - 2] + (x + 5*x^2 + 4*x^3)*deriv(v[i - 2], x) + (5*x^2 + 9*x^3)*deriv(deriv(v[i - 2], x), x) + (2*x^3 + 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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

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. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) =…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.