A360282 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 8\,x^{2} + 5\,n + 20\,n x + 20\,n x^{2} - 2\,n^{2} - 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 8\,x - 24\,n - 16\,n x + 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	5	14	8													27
3	25	86	84	24												219
4	129	522	696	368	64											1779
5	681	3162	5280	4000	1360	160										14643
6	3653	19126	38160	37360	18800	4512	384									121995
7	19825	115502	267204	319480	212240	77280	13888	896								1026315
8	108545	696338	1829072	2576672	2122400	1037120	288512	40448	2048							8701155
9	598417	4191026	12306528	19926144	19559232	11991168	4537344	1001472	112896	4608						74228835
10	3317445	25183710	81680760	149269920	169901760	125241984	60039168	18232320	3283200	304640	10240					636465147
11	18474633	151098246	536142420	1090582680	1410974400	1214564736	705315072	273630720	68471040	10278400	799744	22528				5480354619
12	103274625	905284122	3486753864	7808267280	11309701920	11128962240	7591524864	3590704128	1156320000	243330560	30976000	2052096	49152			47357200851
13	579168825	5416784778	22498512192	54976370592	88087805520	97497148320	76392915456	42661791744	16844914944	4591741440	825876480	90415104	5165056	106496		410468716947
14	3256957317	32372246406	144196223808	381651872784	669975017712	823546467744	728708652672	469402492416	219827781888	73846316544	17306505216	2695999488	256815104	12787712	229376	3567056366187

Technical details

Node id: n-c840071f7c42bf18 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-397aa20e501b57e7

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,       2;
  2:      5,      14,        8;
  3:     25,      86,       84,       24;
  4:    129,     522,      696,      368,       64;
  5:    681,    3162,     5280,     4000,     1360,      160;
  6:   3653,   19126,    38160,    37360,    18800,     4512,     384;
  7:  19825,  115502,   267204,   319480,   212240,    77280,   13888,     896;
  8: 108545,  696338,  1829072,  2576672,  2122400,  1037120,  288512,   40448,   2048;
  9: 598417, 4191026, 12306528, 19926144, 19559232, 11991168, 4537344, 1001472, 112896, 4608;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(8*n^2-20*n+8)*x+(8*n^2-20*n+8)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(8*n^2-16*n+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (8*n^2 - 20*n + 8)*x + (8*n^2 - 20*n + 8)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (8*n^2 - 16*n + 8)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (8*n^2 - 20*n + 8)*x + (8*n^2 - 20*n + 8)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (8*n^2 - 16*n + 8)*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.