Veronese parent of A109446 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + x + 2\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1	1							2
1	2	1							3
2	4	8	5	1					18
3	8	12	6	1					27
4	16	48	56	32	9	1			162
5	32	80	80	40	10	1			243
6	64	256	432	400	220	72	13	1	1458
7	128	448	672	560	280	84	14	1	2187

All 15 rows

Other recurrences

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

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

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

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

Technical details

Node id: n-370e6d9b4c185cdc · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-5b5f75940e0ba52f)
Fingerprint: f6-e891691b80ddc967

Observed on the stored rows ?

Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (irregular)
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;
  1:   2,    1;
  2:   4,    8,    5,    1;
  3:   8,   12,    6,    1;
  4:  16,   48,   56,   32,    9,    1;
  5:  32,   80,   80,   40,   10,    1;
  6:  64,  256,  432,  400,  220,   72,  13,   1;
  7: 128,  448,  672,  560,  280,   84,  14,   1;
  8: 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1;
  9: 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + (2*n + 1)*x + n*x^2)*P[n - 1, x] - (4*x + 4*x^2 + x^3)*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 + x; for(n = 1, N, my(i = n + 1); v[i] = (2 + (2*n + 1)*x + n*x^2)*v[i - 1] - (4*x + 4*x^2 + x^3)*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 veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.?
  - 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 veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.?
  - ↑ Iterated real-rootedness preservers (above)
  - ↑ Parent-row provers of polytool (above)

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.