A357367 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,n x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	18	42	12						72
3	264	1224	840	120					2448
4	5400	46200	58200	19320	1680				130800
5	141840	2157840	4485600	2610720	514080	30240			9940320
6	4551120	120692880	391466880	356680800	119599200	15634080	665280		1009290240
7	172529280	7887841920	38604504960	51818417280	26358393600	5750680320	536215680	17297280	131145880320

All 15 rows

Technical details

Node id: n-4219079737abb529 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b832379b06e619bb

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            1;
  1:            2,              2;
  2:           18,             42,              12;
  3:          264,           1224,             840,              120;
  4:         5400,          46200,           58200,            19320,             1680;
  5:       141840,        2157840,         4485600,          2610720,           514080,           30240;
  6:      4551120,      120692880,       391466880,        356680800,        119599200,        15634080,          665280;
  7:    172529280,     7887841920,     38604504960,      51818417280,      26358393600,      5750680320,       536215680,       17297280;
  8:   7545363840,   590970764160,   4271317868160,    8136500803200,    5897946700800,   1911384034560,    293137004160,    20497276800,    518918400;
  9: 373944211200, 49992123283200, 525947748480000, 1386844870118400, 1377896481561600, 622583766604800, 139703211648000, 15889281408000, 864518054400, 17643225600;

%F Formula

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

%t Mathematica

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

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.?
  - 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A386789 → basis A008277

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.