A159854 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 6\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	6	9															16
1	2	16	42	36														96
2	4	40	148	240	144													576
3	8	96	456	1072	1248	576												3456
4	16	224	1296	3968	6784	6144	2304											20736
5	32	512	3488	13120	29440	39424	29184	9216										124416
6	64	1152	9024	40192	111360	196608	216064	135168	36864									746496
7	128	2560	22656	116480	383488	838656	1218560	1134592	614400	147456								4478976
8	256	5632	55552	323584	1232896	3211264	5791744	7143424	5767168	2752512	589824							26873856
9	512	12288	133632	869376	3760128	11354112	24428544	37453824	40108032	28573696	12189696	2359296						161243136
10	1024	26624	316416	2273280	10997760	37748736	94273536	172621824	230031360	217579520	138674176	53477376	9437184					967458816
11	2048	57344	739328	5812224	31088640	119488512	339542016	722337792	1150550016	1355284480	1147666432	661651456	232783872	37748736				5804752896
12	4096	122880	1708032	14581760	85426176	363331584	1157038080	2802843648	5190451200	7312769024	7716470784	5913968640	3112173568	1006632960	150994944			34828517376
13	8192	262144	3907584	35995648	229179392	1068367872	3767402496	10233839616	21592276992	35387342848	44684017664	42693820416	29880221696	14461960192	4328521728	603979776		208971104256
14	16384	557056	8863744	87621632	602341376	3053453312	11808276480	35537289216	84119912448	157143793664	230917406720	264123711488	230535725056	148444807168	66504884224	18522046464	2415919104	1253826625536

Technical details

Node id: n-26fdd4314504ef0a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-063dae614330e0ae

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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,     6,      9;
  1:   2,    16,     42,     36;
  2:   4,    40,    148,    240,     144;
  3:   8,    96,    456,   1072,    1248,      576;
  4:  16,   224,   1296,   3968,    6784,     6144,     2304;
  5:  32,   512,   3488,  13120,   29440,    39424,    29184,     9216;
  6:  64,  1152,   9024,  40192,  111360,   196608,   216064,   135168,    36864;
  7: 128,  2560,  22656, 116480,  383488,   838656,  1218560,  1134592,   614400,   147456;
  8: 256,  5632,  55552, 323584, 1232896,  3211264,  5791744,  7143424,  5767168,  2752512,   589824;
  9: 512, 12288, 133632, 869376, 3760128, 11354112, 24428544, 37453824, 40108032, 28573696, 12189696, 2359296;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2+6*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.

%t Mathematica

P[0, x_] := 1 + 6*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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

Related ?

Same fingerprint ?

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.