A128540 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

\(P_{2}(x) = 4 + 6\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	4	6	2						12
3	6	15	12	3					36
4	10	35	45	25	5				120
5	16	72	128	112	48	8			384
6	26	143	325	390	260	91	13		1248
7	42	273	756	1155	1050	567	168	21	4032

All 15 rows

Technical details

Node id: n-e6536b8789420799 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-efce8cd29f4f1fd0

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:   2,   1;
  2:   4,   6,    2;
  3:   6,  15,   12,    3;
  4:  10,  35,   45,   25,     5;
  5:  16,  72,  128,  112,    48,     8;
  6:  26, 143,  325,  390,   260,    91,   13;
  7:  42, 273,  756, 1155,  1050,   567,  168,   21;
  8:  68, 510, 1666, 3094,  3570,  2618, 1190,  306,  34;
  9: 110, 935, 3520, 7700, 10780, 10010, 6160, 2420, 550, 55;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) + (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 6*x + 2*x^2.

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 4 + 6*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2] + (n - 2 + (n - 2)*x)*v[i - 1])/(n - 2)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle A127647 * A097806, read by rows.?
  - 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

[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.