A131222 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-288 + 408\,n - 212\,n^{2} + 48\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-180 + 24\,x + 222\,n - 20\,n x - 90\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-52 + 12\,x - x^{2} + 52\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 6\,x + 3\,x^{2}\)

\(P_{3}(x) = 28\,x + 27\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	6	3													9
3	0	28	27	4												59
4	0	180	249	72	5											506
5	0	1488	2610	1100	150	6										5354
6	0	15120	31422	17100	3425	270	7									67344
7	0	182880	431676	283696	74025	8610	441	8								981336
8	0	2570400	6700068	5096784	1621445	246960	18718	672	9							16255056
9	0	41207040	116309520	99482288	36994860	6908454	687960	36624	972	10						301627728
10	0	742452480	2237357808	2107164240	889726000	195359850	23939391	1678320	66150	1350	11					6197745600
11	0	14856307200	47298699360	48291259104	22666660500	5684944980	822268755	71116584	3697650	112200	1815	12				139695068160
12	0	326918592000	1090922821920	1193100350112	612624150380	171824009940	28500638705	2917352592	187475607	7514100	180895	2376	13			3427003088640
13	0	7847004211200	27276742846080	31657283963136	17561846479320	5419987863576	1009405386990	118822073656	9054963918	449193030	14298570	279708	3042	14		90900611562240
14	0	204034563532800	735195556874880	898755755788800	533344882728840	178844651353368	36797929990822	4877453020440	428326036281	25235199990	995317323	25765740	417599	3822	15	2592305376030720

Technical details

Node id: n-27f2c006ebd388c0 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-04811fab6ef6f3bf

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: 0,        2;
  2: 0,        6,         3;
  3: 0,       28,        27,        4;
  4: 0,      180,       249,       72,        5;
  5: 0,     1488,      2610,     1100,      150,       6;
  6: 0,    15120,     31422,    17100,     3425,     270,      7;
  7: 0,   182880,    431676,   283696,    74025,    8610,    441,     8;
  8: 0,  2570400,   6700068,  5096784,  1621445,  246960,  18718,   672,   9;
  9: 0, 41207040, 116309520, 99482288, 36994860, 6908454, 687960, 36624, 972, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(4*n^2-20*n+24)*x)*P(n-3,x) - (13*n^2-52*n+52+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 6*x + 3*x^2, P(3,x) = 28*x + 27*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 6*x + 3*x^2;
P[3, x_] := 28*x + 27*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*P[n - 3, x] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, 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*x; v[3] = 6*x + 3*x^2; v[4] = 28*x + 27*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*v[i - 3] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - 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.