A166961 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-12\,x + 15\,x^{2} - 16\,n x + 14\,n x^{2} + 16\,n^{2} x - 124\,n^{2} x^{2} + 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 18\,x^{2} + 20\,x^{3} + 12\,n x - 32\,n x^{2} + 32\,n x^{3} + 56\,n^{2} x^{2} - 160\,n^{2} x^{3} + 32\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(9\,x^{2} - 18\,x^{3} + 4\,x^{4} + 18\,n x^{2} - 32\,n x^{3} + 8\,n x^{4} + 120\,n^{2} x^{3} - 32\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2\,x^{3} - 4\,x^{4} + 4\,n x^{3} - 8\,n x^{4} + 32\,n^{2} x^{4}\right)\,P_{n-2}'''(x) + \left(-1 + 4\,x + 2\,n - 16\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 4\,x^{2} + 8\,n x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(-2\,x^{2} + 4\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	36	9						46
3	1	236	378	16					631
4	1	1444	8325	2464	25				12259
5	1	8700	147618	142992	12675	36			312022
6	1	52244	2390445	5840800	1687500	57132	49		10028171
7	1	313516	37149714	201789520	139821250	15920712	237748	64	395232525

All 15 rows

Technical details

Node id: n-e09700269a8c9ff3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7938f205f880c3e8

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,       36,          9;
  3: 1,      236,        378,           16;
  4: 1,     1444,       8325,         2464,           25;
  5: 1,     8700,     147618,       142992,        12675,           36;
  6: 1,    52244,    2390445,      5840800,      1687500,        57132,          49;
  7: 1,   313516,   37149714,    201789520,    139821250,     15920712,      237748,        64;
  8: 1,  1881156,  566416053,   6376589856,   9129621375,   2460165192,   129984358,    939264,      81;
  9: 1, 11287004, 8559729810, 191635020304, 520430600025, 280690795692, 35268483824, 961589120, 3578661, 100;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = ((16*n^2-16*n-12)*x+(72*n^3-124*n^2+14*n+15)*x^2)*P(n-2,x) + ((12*n+6)*x+(56*n^2-32*n-18)*x^2+(32*n^3-160*n^2+32*n+20)*x^3)*d/dx P(n-2,x) + ((18*n+9)*x^2+(120*n^2-32*n-18)*x^3-(32*n^2-8*n-4)*x^4)*d^2/dx^2 P(n-2,x) + ((4*n+2)*x^3+(32*n^2-8*n-4)*x^4)*d^3/dx^3 P(n-2,x) + (2*n-1+(16*n^2-16*n+4)*x)*P(n-1,x) + ((8*n-6)*x-(16*n^2-4)*x^2)*d/dx P(n-1,x) + (4*n-2)*x^2*d^2/dx^2 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[Together[(((16*n^2 - 16*n - 12)*x + (72*n^3 - 124*n^2 + 14*n + 15)*x^2)*P[n - 2, x] + ((12*n + 6)*x + (56*n^2 - 32*n - 18)*x^2 + (32*n^3 - 160*n^2 + 32*n + 20)*x^3)*D[P[n - 2, x], x] + ((18*n + 9)*x^2 + (120*n^2 - 32*n - 18)*x^3 - (32*n^2 - 8*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] + ((4*n + 2)*x^3 + (32*n^2 - 8*n - 4)*x^4)*D[P[n - 2, x], {x, 3}] + (2*n - 1 + (16*n^2 - 16*n + 4)*x)*P[n - 1, x] + ((8*n - 6)*x - (16*n^2 - 4)*x^2)*D[P[n - 1, x], x] + (4*n - 2)*x^2*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];
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 + (72*n^3 - 124*n^2 + 14*n + 15)*x^2)*v[i - 2] + ((12*n + 6)*x + (56*n^2 - 32*n - 18)*x^2 + (32*n^3 - 160*n^2 + 32*n + 20)*x^3)*deriv(v[i - 2], x) + ((18*n + 9)*x^2 + (120*n^2 - 32*n - 18)*x^3 - (32*n^2 - 8*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) + ((4*n + 2)*x^3 + (32*n^2 - 8*n - 4)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n - 1 + (16*n^2 - 16*n + 4)*x)*v[i - 1] + ((8*n - 6)*x - (16*n^2 - 4)*x^2)*deriv(v[i - 1], x) + (4*n - 2)*x^2*deriv(deriv(v[i - 1], x), x))/(2*n - 1)); 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) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.