A257622 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,n\right)\,P_{n}(x) = \left(-16 + 16\,x + 8\,x^{2} - 12\,n - 12\,n x + 18\,n x^{2} - 18\,n^{2} x - 27\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 12\,x^{3} - 9\,n x - 36\,n x^{2} + 45\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(8 + 2\,x + 15\,n + 12\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	8														12
2	16	112	48													176
3	64	1104	1656	256												3080
4	256	9392	33120	18784	1280											62832
5	1024	73936	514344	685792	184840	6144										1466080
6	4096	556464	6917904	17830592	11529840	1669392	28672									38536960
7	16384	4075472	85038264	379379648	474224560	170076528	14264152	131072								1127206080
8	65536	29347504	984873216	7086238112	15175185920	10629357168	2298037504	117390016	589824							36321084800
9	262144	209102544	10949263560	121010709792	411101129880	493321355856	211768742136	29198036160	940961448	2621440						1278502184960
10	1048576	1479970736	118274942448	1938114679296	9905412597360	18746211522528	13867577636304	3876229358592	354824827344	7399853680	11534336					48815537971200
11	4194304	10431098320	1251568063704	29611088682240	219052685285760	617680911498336	720627730081392	350484296457216	66624949535040	4171893545680	57371040760	50331648				2009572979814400
12	16777216	73328066736	13047666651360	436675632835552	4541231068450560	18307517877041184	31707620123581248	24410023836054912	8174215923211008	1091689082088880	47841444388320	439968400416	218103808			88730376185651200
13	67108864	514638644432	134546374217448	6268277448390496	89580877867586600	500428203563721312	1231536914132488656	1407470759008558464	750642305345581968	179161755735173200	17237762983073864	538185496869792	3345151188808	939524096		4183003448752128000
14	268435456	3608241873328	1376342060840400	88125227957137216	1699695837437981680	12840544524384926448	43441133427329646624	70373537950427923200	55852885835138117088	21400907540641544080	3739330842363559696	264375683871411648	5964148930308400	25257693113296	4026531840	209707906230773350400

Technical details

Node id: n-bc47a5f111fa8635 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1e1f504198d7902a

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      4,         8;
  2:     16,       112,          48;
  3:     64,      1104,        1656,          256;
  4:    256,      9392,       33120,        18784,         1280;
  5:   1024,     73936,      514344,       685792,       184840,         6144;
  6:   4096,    556464,     6917904,     17830592,     11529840,      1669392,        28672;
  7:  16384,   4075472,    85038264,    379379648,    474224560,    170076528,     14264152,      131072;
  8:  65536,  29347504,   984873216,   7086238112,  15175185920,  10629357168,   2298037504,   117390016,    589824;
  9: 262144, 209102544, 10949263560, 121010709792, 411101129880, 493321355856, 211768742136, 29198036160, 940961448, 2621440;

%F Formula

The row polynomials satisfy (3*n+1)*P(n,x) = -(12*n+16+(18*n^2+12*n-16)*x+(27*n^3+27*n^2-18*n-8)*x^2)*P(n-2,x) - ((9*n+12)*x+(27*n^2+36*n)*x^2-(27*n^2+45*n+12)*x^3)*d/dx P(n-2,x) + (15*n+8+(18*n^2+12*n+2)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*P[n - 2, x] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*D[P[n - 2, x], x] + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x])/(3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*v[i - 2] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*deriv(v[i - 2], x) + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x))/(3*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
    
    ↑ Liu–Wang theorem (above)

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.