A391533 → Pₙ + Pₙ₊₁ → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 1 + 6\,x + 12\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
0	1	6	12	8								27
1	1	7	18	20	8							54
2	1	8	25	38	28	8						108
3	1	9	33	63	66	36	8					216
4	1	10	42	96	129	102	44	8				432
5	1	11	52	138	225	231	146	52	8			864
6	1	12	63	190	363	456	377	198	60	8		1728
7	1	13	75	253	553	819	833	575	258	68	8	3456

All 15 rows

Technical details

Node id: n-f5886681a2578968 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-0dd3ed80d2a0b395

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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,  12,   8;
  1: 1,  7,  18,  20,    8;
  2: 1,  8,  25,  38,   28,    8;
  3: 1,  9,  33,  63,   66,   36,    8;
  4: 1, 10,  42,  96,  129,  102,   44,    8;
  5: 1, 11,  52, 138,  225,  231,  146,   52,    8;
  6: 1, 12,  63, 190,  363,  456,  377,  198,   60,    8;
  7: 1, 13,  75, 253,  553,  819,  833,  575,  258,   68,   8;
  8: 1, 14,  88, 328,  806, 1372, 1652, 1408,  833,  326,  76,  8;
  9: 1, 15, 102, 416, 1134, 2178, 3024, 3060, 2241, 1159, 402, 84, 8;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + 6*x + 12*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 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 + 12*x^2 + 8*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 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 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 shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    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
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A391533 → Pₙ + Pₙ₊₁

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.