A071951 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 13\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-6\,x^{2} + 21\,x^{3}\right)\,P_{n-2}''(x) + \left(-x^{3} + 4\,x^{4}\right)\,P_{n-2}'''(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(5\,x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	4														6
3	4	32	9													45
4	8	208	180	16												412
5	16	1280	2628	640	25											4589
6	32	7744	34416	17472	1750	36										61450
7	64	46592	430416	410624	79800	4032	49									971577
8	128	279808	5269824	8977664	3035600	284256	8232	64								17855576
9	256	1679360	63867456	188921856	105095600	16310016	847896	15360	81							376737881
10	512	10077184	770188032	3891979264	3448058400	836358336	69681920	2213376	26730	100						9028583854
11	1024	60465152	9264930048	79208808448	109522969600	40092254208	5040564144	250376192	5207004	44000	121					243445619941
12	2048	362795008	111315207168	1600647155712	3409452851200	1841587752960	336841604736	24610679808	785512728	11268400	69212	144				7325614899124
13	4096	2176778240	1336598774784	32210836815872	104784596716800	82256297730048	21369735417856	2211925327872	101844037152	2209292800	22770748	104832	169			244276243771269
14	8192	13060685824	16044083048448	646592911917056	3193867334028800	3605654323934208	1308665144199168	187170114764800	11965431336768	368755587200	5678983024	43452864	153790	196		8970346882100338
15	16384	78364147712	192558383124480	12960381052649472	96826321445734400	156036650566238208	78192944238286080	15185525186101248	1313775996808320	55335252044800	1195820019680	13537105920	78986544	219520	225	360764779921482993

Technical details

Node id: n-c3d5489bbfe0c565 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-647f23db98671aaa

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:   2,        4;
   3:   4,       32,         9;
   4:   8,      208,       180,         16;
   5:  16,     1280,      2628,        640,         25;
   6:  32,     7744,     34416,      17472,       1750,        36;
   7:  64,    46592,    430416,     410624,      79800,      4032,       49;
   8: 128,   279808,   5269824,    8977664,    3035600,    284256,     8232,      64;
   9: 256,  1679360,  63867456,  188921856,  105095600,  16310016,   847896,   15360,    81;
  10: 512, 10077184, 770188032, 3891979264, 3448058400, 836358336, 69681920, 2213376, 26730, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-2,x) - (6*x-13*x^2-4*x^3)*d/dx P(n-2,x) - (6*x^2-21*x^3)*d^2/dx^2 P(n-2,x) - (x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2+4*x)*P(n-1,x) + (5*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) = 2 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 2, x] - (6*x - 13*x^2 - 4*x^3)*D[P[n - 2, x], x] - (6*x^2 - 21*x^3)*D[P[n - 2, x], {x, 2}] - (x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2 + 4*x)*P[n - 1, x] + (5*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] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 2] - (6*x - 13*x^2 - 4*x^3)*deriv(v[i - 2], x) - (6*x^2 - 21*x^3)*deriv(deriv(v[i - 2], x), x) - (x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2 + 4*x)*v[i - 1] + (5*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 of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.?
  - 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.