A128540 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 8\,x - 8\,x^{2} + n + 4\,n x + 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-2 - 4\,x + n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 4 + 12\,x + 8\,x^{2}\)

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	2														4
2	4	12	8													24
3	6	30	48	24												108
4	10	70	180	200	80											540
5	16	144	512	896	768	256										2592
6	26	286	1300	3120	4160	2912	832									12636
7	42	546	3024	9240	16800	18144	10752	2688								61236
8	68	1020	6664	24752	57120	83776	76160	39168	8704							297432
9	110	1870	14080	61600	172480	320320	394240	309760	140800	28160						1443420
10	178	3382	28836	145248	478464	1076544	1674624	1777152	1230336	501248	91136					7007148
11	288	6048	57600	328320	1244160	3290112	6193152	8294400	7741440	4792320	1769472	294912				34012224
12	466	10718	112772	717640	3075600	9349824	20668032	33462528	39367680	32806400	18371584	6203392	954368			165101004
13	754	18850	217152	1526096	7298720	25081056	63697920	121026048	171984384	180477440	135888896	69488640	21618688	3088384		801413028
14	1220	32940	412360	3172000	16748160	64201280	184229760	401955840	669926400	848573440	803911680	552181760	259850240	74956800	9994240	3890148120

Technical details

Node id: n-af7f3f008db46a2b · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-04668acf2e74f66d

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,    2;
  2:   4,   12,     8;
  3:   6,   30,    48,    24;
  4:  10,   70,   180,   200,     80;
  5:  16,  144,   512,   896,    768,    256;
  6:  26,  286,  1300,  3120,   4160,   2912,    832;
  7:  42,  546,  3024,  9240,  16800,  18144,  10752,   2688;
  8:  68, 1020,  6664, 24752,  57120,  83776,  76160,  39168,   8704;
  9: 110, 1870, 14080, 61600, 172480, 320320, 394240, 309760, 140800, 28160;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(4*n-8)*x+(4*n-8)*x^2)*P(n-2,x) + (n-2+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 4 + 12*x + 8*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 4 + 12*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*P[n - 2, x] + (n - 2 + (2*n - 4)*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 + 2*x; v[3] = 4 + 12*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*v[i - 2] + (n - 2 + (2*n - 4)*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.