A386789 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(3 + 8\,x - 9\,n - 24\,n x + 6\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

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	8														10
2	9	60	96													165
3	44	416	1280	1280												3020
4	225	2760	12480	24640	17920											58025
5	1182	17832	106176	311808	451584	258048										1146630
6	6321	113204	835520	3252480	7042560	8042496	3784704									23077285
7	34232	709760	6248448	30274560	87183360	149225472	140574720	56229888								470480440
8	187137	4408848	45077760	261233280	938488320	2140250112	3025870848	2424913920	843448320							9683878545
9	1030490	27190280	316595840	2135006720	9189340160	26179112960	49365155840	59416248320	41422684160	12745441280						200797806050
10	5707449	166729068	2177819424	16749587712	83997129216	286996469760	676614242304	1086867505152	1138486542336	702273814528	193730707456					4188066254405
11	31760676	1017599904	14734437120	127269192960	728615946240	2903011098624	8213944467456	16505056788480	23082573496320	21398437232640	11835185037312	2958796259328				87768673317060
12	177435297	6186624600	98346376128	942498997440	6064641623040	27603519111168	91127913676800	219863163666432	384767347261440	476335376957440	395985566040064	198485915729920	45368209309696			1846648862809465
13	994551222	37489038888	649055211840	6833851960320	48820403681280	249892157104128	942878039212032	2655279596175360	5581162803363840	8647971129262080	9601919760728064	7234484453769216	3315369141862400	697972450918400		38983271326839070
14	5590402785	226534642020	4242869055360	48682710535680	382299349386240	2173553537126400	9226583862312960	29706999655956480	72903014415728640	135704364877086720	188614279472087040	189817347165388800	130757660403302400	55189678797619200	10768717814169600	825297657054800325

Technical details

Node id: n-612d53fdc9f3ca39 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-bf81431d3bf08044

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,        8;
  2:       9,       60,        96;
  3:      44,      416,      1280,       1280;
  4:     225,     2760,     12480,      24640,      17920;
  5:    1182,    17832,    106176,     311808,     451584,      258048;
  6:    6321,   113204,    835520,    3252480,    7042560,     8042496,     3784704;
  7:   34232,   709760,   6248448,   30274560,   87183360,   149225472,   140574720,    56229888;
  8:  187137,  4408848,  45077760,  261233280,  938488320,  2140250112,  3025870848,  2424913920,   843448320;
  9: 1030490, 27190280, 316595840, 2135006720, 9189340160, 26179112960, 49365155840, 59416248320, 41422684160, 12745441280;

%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+(16*n^2-24*n+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n)*P[n - 2, x] + (6*n^2 - 9*n + 3 + (16*n^2 - 24*n + 8)*x)*P[n - 1, 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n)*v[i - 2] + (6*n^2 - 9*n + 3 + (16*n^2 - 24*n + 8)*x)*v[i - 1])/(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 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? 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

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.