A386789 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x - 9\,n - 6\,n x + 6\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 6\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	9	21	6													36
3	44	204	140	20												408
4	225	1925	2425	805	70											5450
5	1182	17982	37380	21756	4284	252										82836
6	6321	167629	543704	495390	166110	21714	924									1401792
7	34232	1565048	7659624	10281432	5229840	1141008	106392	3432								26021008
8	187137	14657013	105935463	201798135	146278440	47405358	7270263	508365	12870							524053044
9	1030490	137764890	1449371000	3821772680	3797113320	1715673960	384984600	43786600	2382380	48620						11353928540
10	5707449	1299653861	19705976026	70680546508	93752379292	56764039332	17485159692	2876796780	252381558	10992982	184756					262833818236
11	31760676	12303590532	267001378644	1286221703052	2235253740720	1765199772336	717100327944	160087242744	20136556440	1404313560	50085672	705432				6464791177752
12	177435297	116849655013	3611517736853	23146574620169	51971886792826	52499763328012	27367337955958	7962445313394	1346744589762	133703760190	7590977394	225797026	2704156			168164820666050
13	994551222	1112958527254	48821255405468	413325207775844	1186381601945540	1510767855689092	990707265716168	365422957667768	79814306551980	10578074527020	849954088984	40051046504	1008858200	10400600		4607823502751644
14	5590402785	10628074990365	660046079327340	7341089165507370	26714518593347130	42404690383714230	34461731186333880	15794602187492700	4332869635092570	736143781903530	78495672495240	5209218514500	207012799500	4473000900	40116600	132540241095038640

Technical details

Node id: n-b882a6077366aded · 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:       9,        21,          6;
  3:      44,       204,        140,         20;
  4:     225,      1925,       2425,        805,         70;
  5:    1182,     17982,      37380,      21756,       4284,        252;
  6:    6321,    167629,     543704,     495390,     166110,      21714,       924;
  7:   34232,   1565048,    7659624,   10281432,    5229840,    1141008,    106392,     3432;
  8:  187137,  14657013,  105935463,  201798135,  146278440,   47405358,   7270263,   508365,   12870;
  9: 1030490, 137764890, 1449371000, 3821772680, 3797113320, 1715673960, 384984600, 43786600, 2382380, 48620;

%F Formula

The row polynomials satisfy (n^2-n)*P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n^2-9*n+3+(4*n^2-6*n+2)*x)*P(n-1,x) + (4*n^2-6*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[Together[(-(n^2 - 2*n)*P[n - 2, x] + (6*n^2 - 9*n + 3 + (4*n^2 - 6*n + 2)*x)*P[n - 1, x] + (4*n^2 - 6*n + 2)*x*D[P[n - 1, x], x])/(n^2 - n)]];
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^2 - 9*n + 3 + (4*n^2 - 6*n + 2)*x)*v[i - 1] + (4*n^2 - 6*n + 2)*x*deriv(v[i - 1], x))/(n^2 - n)); 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)*binomial(n + k, k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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 ?

A357367 → 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.