A204420 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 48\,x + 27\,x^{2} + 176\,n - 152\,n x - 72\,n x^{2} - 204\,n^{2} + 128\,n^{2} x + 36\,n^{2} x^{2} + 96\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 12\,x^{3} + 152\,n x^{2} - 32\,n x^{3} - 128\,n^{2} x^{2} + 16\,n^{2} x^{3} + 32\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(6 - 4\,x - 16\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x^{2} - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	24	27						51
3	0	480	810	240					1530
4	0	20160	41580	20160	2625				84525
5	0	1451520	3402000	2116800	472500	34020			7476840
6	0	159667200	410145120	299376000	88357500	11226600	509355		969281775
7	0	24908083200	68652904320	56181565440	19864845000	3405402000	278107830	8648640	173299556430

All 15 rows

Technical details

Node id: n-7df447b935390019 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-751d7e1b87540300

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,                4;
  2: 0,               24,               27;
  3: 0,              480,              810,              240;
  4: 0,            20160,            41580,            20160,             2625;
  5: 0,          1451520,          3402000,          2116800,           472500,           34020;
  6: 0,        159667200,        410145120,        299376000,         88357500,        11226600,         509355;
  7: 0,      24908083200,      68652904320,      56181565440,      19864845000,      3405402000,      278107830,       8648640;
  8: 0,    5230697472000,   15257757715200,   13628872857600,    5488372890000,   1144215072000,   127929601800,    7264857600,    164189025;
  9: 0, 1422749712384000, 4350184276838400, 4168176761548800, 1854854361360000, 445581867931200, 61272717105600, 4816600588800, 200967366600, 3445942500;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(32*n^3-128*n^2+152*n-48)*x-(36*n^2-72*n+27)*x^2)*P(n-2,x) + ((32*n^3-128*n^2+152*n-48)*x^2+(16*n^2-32*n+12)*x^3)*d/dx P(n-2,x) + (8*n^2-16*n+6+(8*n-4)*x)*P(n-1,x) - (8*n-4)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*P[n - 1, x] - (8*n - 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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*v[i - 1] - (8*n - 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.