A136158 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	8	18	9													35
3	32	96	90	27												245
4	128	480	648	378	81											1715
5	512	2304	4032	3456	1458	243										12005
6	2048	10752	23040	25920	16200	5346	729									84035
7	8192	49152	124416	172800	142560	69984	18954	2187								588245
8	32768	221184	645120	1064448	1088640	707616	285768	65610	6561							4117715
9	131072	983040	3244032	6193152	7547904	6096384	3265920	1119744	223074	19683						28824005
10	524288	4325376	15925248	34504704	48771072	47029248	31352832	14276736	4251528	747954	59049					201768035
11	2097152	18874368	76677120	185794560	298598400	334430208	266499072	151165440	59836320	15746400	2480058	177147				1412376245
12	8388608	81788928	363331584	973209600	1751777280	2233516032	2069286912	1404158976	692841600	242494560	57159432	8148762	531441			9886633715
13	33554432	352321536	1698693120	4982833152	9926737920	14189395968	14977695744	11824496640	6983843328	3048503040	956121408	204073344	26572050	1594323		69206436005
14	134217728	1509949440	7851737088	25027411968	54655451136	86537797632	102478970880	92231073792	63408863232	33145542144	12969994752	3684657600	718508232	86093442	4782969	484445052035

Technical details

Node id: n-4110aee7a83ba37b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-c4ad8dc4ca5d500a

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:      2,      3;
  2:      8,     18,       9;
  3:     32,     96,      90,      27;
  4:    128,    480,     648,     378,      81;
  5:    512,   2304,    4032,    3456,    1458,     243;
  6:   2048,  10752,   23040,   25920,   16200,    5346,     729;
  7:   8192,  49152,  124416,  172800,  142560,   69984,   18954,    2187;
  8:  32768, 221184,  645120, 1064448, 1088640,  707616,  285768,   65610,   6561;
  9: 131072, 983040, 3244032, 6193152, 7547904, 6096384, 3265920, 1119744, 223074, 19683;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*x)*P[n - 1, x])/(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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*x)*v[i - 1])/(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 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 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
  - 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
[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.