A131222 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-288 + 408\,n - 212\,n^{2} + 48\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-180 + 48\,x + 222\,n - 40\,n x - 90\,n^{2} + 8\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-48\,x^{2} + 40\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-3}'(x) + \left(-52 + 24\,x + 9\,x^{2} + 52\,n - 12\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} + 4\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-9 + 4\,x + 6\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 12\,x + 9\,x^{2}\)

\(P_{3}(x) = 56\,x + 81\,x^{2} + 16\,x^{3}\)

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	56	81	16					153
4	0	360	747	288	25				1420
5	0	2976	7830	4400	750	36			15992
6	0	30240	94266	68400	17125	1620	49		211700
7	0	365760	1295028	1134784	370125	51660	3087	64	3220508

All 15 rows

Technical details

Node id: n-3ca3dd82909bb758 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e79a6bc76e422282

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,       56,        81,        16;
  4: 0,      360,       747,       288,        25;
  5: 0,     2976,      7830,      4400,       750,       36;
  6: 0,    30240,     94266,     68400,     17125,     1620,      49;
  7: 0,   365760,   1295028,   1134784,    370125,    51660,    3087,     64;
  8: 0,  5140800,  20100204,  20387136,   8107225,  1481760,  131026,   5376,   81;
  9: 0, 82414080, 348928560, 397929152, 184974300, 41450724, 4815720, 292992, 8748, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(8*n^2-40*n+48)*x)*P(n-3,x) - (8*n^2-40*n+48)*x^2*d/dx P(n-3,x) - (13*n^2-52*n+52+(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 >= 4, with P(0,x) = 1, P(1,x) = 4*x, P(2,x) = 12*x + 9*x^2, P(3,x) = 56*x + 81*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[2, x_] := 12*x + 9*x^2;
P[3, x_] := 56*x + 81*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (8*n^2 - 40*n + 48)*x)*P[n - 3, x] - (8*n^2 - 40*n + 48)*x^2*D[P[n - 3, x], x] - (13*n^2 - 52*n + 52 + (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; v[3] = 12*x + 9*x^2; v[4] = 56*x + 81*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (8*n^2 - 40*n + 48)*x)*v[i - 3] - (8*n^2 - 40*n + 48)*x^2*deriv(v[i - 3], x) - (13*n^2 - 52*n + 52 + (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 Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - 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.