A142963 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12\,x^{2} + 16\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 8\,x^{3} - 4\,n x^{2} + 16\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 - 4\,x + 8\,n x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	20	12						33
3	1	60	216	32					309
4	1	148	1548	1856	80				3633
5	1	332	7752	28064	14320	192			50661
6	1	708	32220	256960	422400	104256	448		816993
7	1	1468	120024	1801120	6608800	5694336	732928	1024	14959701

All 15 rows

Technical details

Node id: n-711a16891064929c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-90bc0c24022645a4

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: 1,   20,      12;
  3: 1,   60,     216,       32;
  4: 1,  148,    1548,     1856,        80;
  5: 1,  332,    7752,    28064,     14320,        192;
  6: 1,  708,   32220,   256960,    422400,     104256,        448;
  7: 1, 1468,  120024,  1801120,   6608800,    5694336,     732928,      1024;
  8: 1, 2996,  417324, 10725184,  73569200,  145193856,   71564416,   5033984,     2304;
  9: 1, 6060, 1386792, 57367968, 662788560, 2460257856, 2872450560, 858150912, 34000128, 5120;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-16*n-12)*x^2*P(n-2,x) - ((4*n+2)*x^2-(16*n+8)*x^3)*d/dx P(n-2,x) + (1+(8*n-4)*x)*P(n-1,x) + (x-4*x^2)*d/dx 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[-(16*n^2 - 16*n - 12)*x^2*P[n - 2, x] - ((4*n + 2)*x^2 - (16*n + 8)*x^3)*D[P[n - 2, x], x] + (1 + (8*n - 4)*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], 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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 16*n - 12)*x^2*v[i - 2] - ((4*n + 2)*x^2 - (16*n + 8)*x^3)*deriv(v[i - 2], x) + (1 + (8*n - 4)*x)*v[i - 1] + (x - 4*x^2)*deriv(v[i - 1], 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
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 read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…?
  - 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.