A360282 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 16\,x - 32\,x^{2} + 5\,n + 40\,n x + 80\,n x^{2} - 2\,n^{2} - 16\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 16\,x - 24\,n - 32\,n x + 12\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	5	28	32													65
3	25	172	336	192												725
4	129	1044	2784	2944	1024											7925
5	681	6324	21120	32000	21760	5120										87005
6	3653	38252	152640	298880	300800	144384	24576									963185
7	19825	231004	1068816	2555840	3395840	2472960	888832	114688								10747805
8	108545	1392676	7316288	20613376	33958400	33187840	18464768	5177344	524288							120743525
9	598417	8382052	49226112	159409152	312947712	383717376	290390016	128188416	28901376	2359296						1364119925
10	3317445	50367420	326723040	1194159360	2718428160	4007743488	3842506752	2333736960	840499200	155975680	10485760					15483943265
11	18474633	302196492	2144569680	8724661440	22575590400	38866071552	45140164608	35024732160	17528586240	5262540800	818937856	46137344				176452663205
12	103274625	1810568244	13947015456	62466138240	180955230720	356126791680	485857591296	459610128384	296017920000	124585246720	31719424000	4202692608	201326592			2017603348565
13	579168825	10833569556	89994048768	439810964736	1409404888320	3119908746240	4889146589184	5460709343232	4312298225664	2350971617280	845697515520	185170132992	21156069376	872415232		23136553294925
14	3256957317	64744492812	576784895232	3053214982272	10719600283392	26353486967808	46637353771008	60083519029248	56275912163328	37809314070528	17721861341184	5521406951424	1051914665984	104756936704	3758096384	265980885604625

Technical details

Node id: n-d8c5dd46f02744fc · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-0533d0ada67c5eb8

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:      1,       4;
  2:      5,      28,       32;
  3:     25,     172,      336,       192;
  4:    129,    1044,     2784,      2944,      1024;
  5:    681,    6324,    21120,     32000,     21760,      5120;
  6:   3653,   38252,   152640,    298880,    300800,    144384,     24576;
  7:  19825,  231004,  1068816,   2555840,   3395840,   2472960,    888832,    114688;
  8: 108545, 1392676,  7316288,  20613376,  33958400,  33187840,  18464768,   5177344,   524288;
  9: 598417, 8382052, 49226112, 159409152, 312947712, 383717376, 290390016, 128188416, 28901376, 2359296;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(16*n^2-40*n+16)*x+(32*n^2-80*n+32)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(16*n^2-32*n+16)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (16*n^2 - 40*n + 16)*x + (32*n^2 - 80*n + 32)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (16*n^2 - 32*n + 16)*x)*P[n - 1, x])/(2*n^2 - 3*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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (16*n^2 - 40*n + 16)*x + (32*n^2 - 80*n + 32)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (16*n^2 - 32*n + 16)*x)*v[i - 1])/(2*n^2 - 3*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) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
  - 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

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.