A134239 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} - 4\,n - 12\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 12\,x + 4\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 6\,x\)

\(P_{2}(x) = 18 + 45\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	6														12
2	18	45	27													90
3	48	192	252	108												600
4	120	660	1350	1215	405											3750
5	288	2016	5616	7776	5346	1458										22500
6	672	5712	20160	37800	39690	22113	5103									131250
7	1536	15360	65664	155520	220320	186624	87480	17496								750000
8	3456	39744	199584	571536	1020600	1163484	826686	334611	59049							4218750
9	7680	99840	576000	1935360	4173120	5987520	5715360	3499200	1246590	196830						23437500
10	16896	244992	1596672	6158592	15567552	26943840	32332608	26558928	14289858	4546773	649539					128906250
11	36864	589824	4285440	18662400	54120960	109734912	158723712	163762560	118098000	56687040	16297524	2125764				703125000
12	79872	1397760	11202048	54362880	177914880	413652096	700539840	870670944	788107320	506640420	219544182	57572775	6908733			3808593750
13	172032	3268608	28643328	153280512	558835200	1465744896	2845269504	4138573824	4510398816	3637418400	2109702672	833299488	200884698	22320522		20507812500
14	368640	7557120	71884800	420526080	1690191360	4937137920	10808328960	18013881600	22967699040	22292178480	16212493440	8566828920	3108929850	693530505	71744535	109863281250

Technical details

Node id: n-fd95b73866112a83 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b821bbf6b18e27b2

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    6,     6;
  2:   18,    45,     27;
  3:   48,   192,    252,     108;
  4:  120,   660,   1350,    1215,     405;
  5:  288,  2016,   5616,    7776,    5346,    1458;
  6:  672,  5712,  20160,   37800,   39690,   22113,    5103;
  7: 1536, 15360,  65664,  155520,  220320,  186624,   87480,   17496;
  8: 3456, 39744, 199584,  571536, 1020600, 1163484,  826686,  334611,   59049;
  9: 7680, 99840, 576000, 1935360, 4173120, 5987520, 5715360, 3499200, 1246590, 196830;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(12*n-24)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-8+(6*n-12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 6*x, P(2,x) = 18 + 45*x + 27*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[2, x_] := 18 + 45*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 8 + (6*n - 12)*x)*P[n - 1, x])/(n - 2)]];
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] = 6 + 6*x; v[3] = 18 + 45*x + 27*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 8 + (6*n - 12)*x)*v[i - 1])/(n - 2)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A127899(unsigned) * A007318.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.