A269939 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,n + n^{2}\right)\,P_{n}(x) = \left(8\,x + 288\,x^{2} - 4\,n x - 498\,n x^{2} + 317\,n^{2} x^{2} - 88\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x - 26\,x^{2} - 81\,x^{3} + 19\,n x^{2} + 36\,n x^{3} - 4\,n^{2} x^{2} + 39\,n^{2} x^{3} - 26\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-7\,x^{2} - 70\,x^{3} - 63\,x^{4} + 5\,n x^{2} + 83\,n x^{3} + 78\,n x^{4} - n^{2} x^{2} - 32\,n^{2} x^{3} - 31\,n^{2} x^{4} + 4\,n^{3} x^{3} + 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-6 - 36\,x + 5\,n + 60\,n x - n^{2} - 28\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(16\,x + 72\,x^{2} - 11\,n x - 84\,n x^{2} + 2\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\,x\)

\(P_{3}(x) = 4\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	4							4
3	0	4	27						31
4	0	4	90	240					334
5	0	4	225	1680	2625				4534
6	0	4	504	7840	31500	34020			73868
7	0	4	1071	30688	236250	623700	509355		1401068
8	0	4	2214	109200	1424500	6860700	13243230	8648640	30288488

All 15 rows

Technical details

Node id: n-35cc86536da6692a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-be569a6fd03aad81

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0, 4;
   3: 0, 4,   27;
   4: 0, 4,   90,     240;
   5: 0, 4,  225,    1680,     2625;
   6: 0, 4,  504,    7840,    31500,     34020;
   7: 0, 4, 1071,   30688,   236250,    623700,     509355;
   8: 0, 4, 2214,  109200,  1424500,   6860700,   13243230,    8648640;
   9: 0, 4, 4509,  366960,  7574875,  58918860,  200855655,  302702400,  164189025;
  10: 0, 4, 9108, 1189056, 37180000, 436395960, 2327865540, 6054048000, 7443235800, 3445942500;

%F Formula

The row polynomials satisfy (n^2-6*n+9)*P(n,x) = -((4*n-8)*x-(9*n^4-88*n^3+317*n^2-498*n+288)*x^2)*P(n-2,x) - (x+(4*n^2-19*n+26)*x^2-(4*n^4-26*n^3+39*n^2+36*n-81)*x^3)*d/dx P(n-2,x) - ((n^2-5*n+7)*x^2-(4*n^3-32*n^2+83*n-70)*x^3-(4*n^3-31*n^2+78*n-63)*x^4)*d^2/dx^2 P(n-2,x) - (n^2-5*n+6-(4*n^3-28*n^2+60*n-36)*x)*P(n-1,x) + ((2*n^2-11*n+16)*x-(4*n^3-32*n^2+84*n-72)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 4*x + 27*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4*x;
P[3, x_] := 4*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*P[n - 2, x] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*D[P[n - 2, x], x] - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*D[P[n - 2, x], {x, 2}] - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*P[n - 1, x] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*D[P[n - 1, x], x])/(n^2 - 6*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 4*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*v[i - 2] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*deriv(v[i - 2], x) - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*deriv(deriv(v[i - 2], x), x) - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*v[i - 1] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*deriv(v[i - 1], x))/(n^2 - 6*n + 9)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.