A144697 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-12\,x + 50\,x^{2} - 16\,n x + 93\,n x^{2} - 4\,n^{2} x + 52\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 37\,x^{2} - 50\,x^{3} - 2\,n x + 24\,n x^{2} - 21\,n x^{3} + 4\,n^{2} x^{2} + 10\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{2} + 25\,x^{3} - 22\,x^{4} - n x^{2} + 20\,n x^{3} - 19\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2 + 16\,x + n + 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(5\,x - 24\,x^{2} + 2\,n x - 20\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 12\,x\)

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	12														13
2	1	40	81													122
3	1	100	603	432												1136
4	1	224	2934	6016	2025											11200
5	1	476	11826	50144	47725	8748										118920
6	1	984	42975	326720	630375	327384	35721									1364160
7	1	2004	146637	1841680	6214875	6503004	2032471	139968								16880640
8	1	4048	480492	9452224	51217750	92714544	58483852	11738368	531441							224622720
9	1	8140	1532556	45496768	374241550	1072562184	1166555740	475842304	64208457	1968300						3202416000
10	1	16328	4799133	209232512	2511006050	10746635760	18385024322	12948726272	3589065693	336761800	7144929					48738412800
11	1	32708	14838255	930779760	15824288250	97022452968	245714092974	271681461120	130630981365	25522344500	1708521331	25509168				789075302400
12	1	65472	45471474	4039668480	95119218375	810004468608	2908523699676	4740913231872	3582606776895	1222860344000	173203918866	8439236352	89813529			13545845913600
13	1	131004	138476790	17209570208	551339875875	6366715230708	31384726720452	72117298732800	80245207464759	43189402746500	10783209204966	1131905725344	40785102085	312487308		245828251468800
14	1	262072	419851755	72284814272	3106268774225	47727444439368	315016962302931	986861351109120	1543667973061131	1224439286376200	484429542517481	90580329473472	7172292007075	193579082872	1076168025	4703268810240000

Technical details

Node id: n-4446570c0ecf5961 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-eb8c27c5cf228539

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,   12;
  2: 1,   40,      81;
  3: 1,  100,     603,      432;
  4: 1,  224,    2934,     6016,      2025;
  5: 1,  476,   11826,    50144,     47725,       8748;
  6: 1,  984,   42975,   326720,    630375,     327384,      35721;
  7: 1, 2004,  146637,  1841680,   6214875,    6503004,    2032471,    139968;
  8: 1, 4048,  480492,  9452224,  51217750,   92714544,   58483852,  11738368,   531441;
  9: 1, 8140, 1532556, 45496768, 374241550, 1072562184, 1166555740, 475842304, 64208457, 1968300;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -((4*n^2+16*n+12)*x-(9*n^3+52*n^2+93*n+50)*x^2)*P(n-2,x) - ((2*n+6)*x-(4*n^2+24*n+37)*x^2-(4*n^3+10*n^2-21*n-50)*x^3)*d/dx P(n-2,x) - ((n+3)*x^2-(4*n^2+20*n+25)*x^3+(4*n^2+19*n+22)*x^4)*d^2/dx^2 P(n-2,x) + (n+2+(4*n^2+16*n+16)*x)*P(n-1,x) + ((2*n+5)*x-(4*n^2+20*n+24)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 16*n + 12)*x - (9*n^3 + 52*n^2 + 93*n + 50)*x^2)*P[n - 2, x] - ((2*n + 6)*x - (4*n^2 + 24*n + 37)*x^2 - (4*n^3 + 10*n^2 - 21*n - 50)*x^3)*D[P[n - 2, x], x] - ((n + 3)*x^2 - (4*n^2 + 20*n + 25)*x^3 + (4*n^2 + 19*n + 22)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 2 + (4*n^2 + 16*n + 16)*x)*P[n - 1, x] + ((2*n + 5)*x - (4*n^2 + 20*n + 24)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 + 16*n + 12)*x - (9*n^3 + 52*n^2 + 93*n + 50)*x^2)*v[i - 2] - ((2*n + 6)*x - (4*n^2 + 24*n + 37)*x^2 - (4*n^3 + 10*n^2 - 21*n - 50)*x^3)*deriv(v[i - 2], x) - ((n + 3)*x^2 - (4*n^2 + 20*n + 25)*x^3 + (4*n^2 + 19*n + 22)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 2 + (4*n^2 + 16*n + 16)*x)*v[i - 1] + ((2*n + 5)*x - (4*n^2 + 20*n + 24)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 Triangle of 3-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.