A103633 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 6\,x - 7\,n - 4\,n x\right)\,P_{n-1}(x) + \left(12 + 14\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	12	14	4						30
3	36	66	40	8					150
4	144	336	292	112	16				900
5	432	1296	1548	920	272	32			4500
6	1728	6048	8784	6776	2928	672	64		27000
7	5184	21600	38448	37896	22336	7872	1536	128	135000

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(12 + 14\,x + 4\,x^{2}\right)\,P_{n-2}(x)\)

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

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

Technical details

Node id: n-17ae8aea8bf71051 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-b3c91bcca268a73d)
Fingerprint: f6-a7854389020f6acb

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:     3,      2;
  2:    12,     14,      4;
  3:    36,     66,     40,       8;
  4:   144,    336,    292,     112,     16;
  5:   432,   1296,   1548,     920,    272,     32;
  6:  1728,   6048,   8784,    6776,   2928,    672,     64;
  7:  5184,  21600,  38448,   37896,  22336,   7872,   1536,   128;
  8: 20736,  96768, 196992,  228480, 165136,  76160,  21888,  3584,  256;
  9: 62208, 331776, 784512, 1079424, 952368, 558752, 217984, 54528, 7936, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(7*n-10+(4*n-6)*x)*P(n-1,x) + (12+14*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(7*n - 10 + (4*n - 6)*x)*P[n - 1, x] + (12 + 14*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(7*n - 10 + (4*n - 6)*x)*v[i - 1] + (12 + 14*x + 4*x^2)*deriv(v[i - 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]
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), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - ↑ Geometric basis map (elementary) (above)

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.