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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	68	510	1666	3094	3570	2618	1190	306	34							13056
9	110	935	3520	7700	10780	10010	6160	2420	550	55						42240
10	178	1691	7209	18156	29904	33642	26166	13884	4806	979	89					136704
11	288	3024	14400	41040	77760	102816	96768	64800	30240	9360	1728	144				442368
12	466	5359	28193	89705	192225	292182	322938	261426	153780	64075	17941	3029	233			1431552
13	754	9425	54288	190762	456170	783783	995280	945516	671814	352495	132704	33930	5278	377		4632576
14	1220	16470	103090	396500	1046760	2006290	2878590	3140280	2616900	1657370	785070	269620	63440	9150	610	14991360

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.