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	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

All 15 rows

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.