A142075 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	16	12													29
4	1	44	132	32												209
5	1	104	792	832	80											1809
6	1	228	3624	9664	4560	192										18269
7	1	480	14292	77312	95280	23040	448									210853
8	1	988	51516	499808	1249520	824256	110656	1024								2737769
9	1	2008	175296	2823488	12495200	16940928	6544384	514048	2304							39497657
10	1	4052	574080	14566144	104828320	251587968	203926016	48988160	2333952	5120						626813813
11	1	8144	1831644	70511616	779049120	3019055616	4362675072	2256371712	351675648	10424320	11264					10851614157
12	1	16332	5739252	326005920	5305477920	31202358912	72805504128	67910117376	23472426240	2448747520	45990912	24576				203522409089
13	1	32712	17756712	1457070400	33862522800	289079329536	1019277057792	1541756424192	975240656640	233131264000	16667633664	200982528	53248			4110690784225
14	1	65476	54447768	6349145152	205739374000	2465906469696	12531086889216	28642484318208	29590877636352	13167319936000	2234899093504	111509028864	871616512	114688		88957098135437
15	1	131008	165896868	27138909184	1203298361680	19733183794176	139501644088128	458279749091328	717437026738944	526218234511360	169424409324544	20842682253312	736139702272	3756261376	245760	2053407429309941

Technical details

Node id: n-498249210f3279f0 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f712c08c0fc317d2

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: 1,    4;
   3: 1,   16,     12;
   4: 1,   44,    132,       32;
   5: 1,  104,    792,      832,        80;
   6: 1,  228,   3624,     9664,      4560,       192;
   7: 1,  480,  14292,    77312,     95280,     23040,       448;
   8: 1,  988,  51516,   499808,   1249520,    824256,    110656,     1024;
   9: 1, 2008, 175296,  2823488,  12495200,  16940928,   6544384,   514048,    2304;
  10: 1, 4052, 574080, 14566144, 104828320, 251587968, 203926016, 48988160, 2333952, 5120;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n)*x^2*P[n - 2, x] - (2*n*x^2 - 4*n*x^3)*D[P[n - 2, x], x] + (1 + (4*n - 4)*x)*P[n - 1, x] + (x - 2*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] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 8*n)*x^2*v[i - 2] - (2*n*x^2 - 4*n*x^3)*deriv(v[i - 2], x) + (1 + (4*n - 4)*x)*v[i - 1] + (x - 2*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 T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A156365 → θ+1

References

[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.