A090452 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,x + 2\,n + 6\,n x + 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(12\,x + 24\,x^{2} + 12\,x^{3} + 2\,n + 14\,n x + 36\,n x^{2} + 24\,n x^{3} + 3\,n^{2} + 15\,n^{2} x + 12\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4} + 3\,n x + 14\,n x^{2} + 19\,n x^{3} + 8\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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	6	6													13
2	1	14	48	60	25											148
3	1	24	153	420	570	378	98									1644
4	1	36	354	1584	3855	5460	4508	2016	378							18192
5	1	50	690	4440	16175	36498	53060	49920	29430	9900	1452					201616
6	1	66	1206	10400	51825	164898	353248	522648	536625	376200	172062	46332	5577			2241088
7	1	84	1953	21560	139125	583128	1685257	3475944	5208210	5694150	4502652	2510508	936936	210210	21450	24991168

All 15 rows

Technical details

Node id: n-d4944ce4c8038cb8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9c33180785a5ceeb

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,   6,    6;
  2: 1,  14,   48,    60,     25;
  3: 1,  24,  153,   420,    570,     378,       98;
  4: 1,  36,  354,  1584,   3855,    5460,     4508,     2016,       378;
  5: 1,  50,  690,  4440,  16175,   36498,    53060,    49920,     29430,      9900,      1452;
  6: 1,  66, 1206, 10400,  51825,  164898,   353248,   522648,    536625,    376200,    172062,     46332,      5577;
  7: 1,  84, 1953, 21560, 139125,  583128,  1685257,  3475944,   5208210,   5694150,   4502652,   2510508,    936936,    210210,     21450;
  8: 1, 104, 2988, 40880, 329140, 1739304,  6429388, 17290304,  34647435,  52410600,  60082792,  51939888,  33339306,  15415400,   4856280,   933504,    82654;
  9: 1, 126, 4374, 72384, 707760, 4572288, 20855184, 70056864, 178145163, 348956850, 531794142, 632681712, 585946998, 418256748, 225735120, 89107200, 24277734, 4081428, 319124;

%F Formula

The row polynomials satisfy (n^3+3*n^2+2*n+(4*n^2+6*n-4)*x)*P(n,x) = (n^3+3*n^2+2*n+(n^3+15*n^2+14*n+12)*x+(12*n^2+36*n+24)*x^2+(24*n+12)*x^3)*P(n-1,x) + ((2*n^2+3*n+4)*x+(4*n^2+14*n+12)*x^2+(2*n^2+19*n+12)*x^3+(8*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*P[n - 1, x] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*v[i - 1] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*x)); 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 Scaled array A078740 ((3,2)-Stirling2).?
  - 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.