basis-map coefficients of A371080, shifted → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 24\,x + 9\,x^{2} + 36\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} + 4\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-9 + 4\,x + 6\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	12	9						21
3	0	72	81	16					169
4	0	648	891	288	25				1852
5	0	7776	12150	5040	750	36			25752
6	0	116640	199746	97200	19125	1620	49		434380
7	0	2099520	3857868	2104704	496125	56700	3087	64	8618068

All 15 rows

Technical details

Node id: n-316a90fb75a750e7 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-318d7316a46a53d1

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,         12,          9;
  3: 0,         72,         81,         16;
  4: 0,        648,        891,        288,        25;
  5: 0,       7776,      12150,       5040,       750,       36;
  6: 0,     116640,     199746,      97200,     19125,     1620,      49;
  7: 0,    2099520,    3857868,    2104704,    496125,    56700,    3087,     64;
  8: 0,   44089920,   85739148,   51057216,  13707225,  1905120,  142002,   5376,   81;
  9: 0, 1058158080, 2156941872, 1377798336, 408750300, 65461284, 6001128, 314496, 8748, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-36*n+36+(12*n-24)*x-9*x^2)*P(n-2,x) + ((12*n-24)*x^2+4*x^3)*d/dx P(n-2,x) + (6*n-9+4*x)*P(n-1,x) - 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[-(9*n^2 - 36*n + 36 + (12*n - 24)*x - 9*x^2)*P[n - 2, x] + ((12*n - 24)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (6*n - 9 + 4*x)*P[n - 1, 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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 36*n + 36 + (12*n - 24)*x - 9*x^2)*v[i - 2] + ((12*n - 24)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (6*n - 9 + 4*x)*v[i - 1] - 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 basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
  - Parent-row provers of polytool 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.