A201637 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-5 + 2\,n\right)\,P_{n}(x) = \left(-72\,x - 870\,x^{2} + 72\,n x + 1178\,n x^{2} - 16\,n^{2} x - 512\,n^{2} x^{2} + 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 50\,x^{2} - 600\,x^{3} - 4\,n x - 56\,n x^{2} + 740\,n x^{3} + 16\,n^{2} x^{2} - 280\,n^{2} x^{3} + 32\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(3\,x^{2} + 62\,x^{3} - 65\,x^{4} - 2\,n x^{2} - 64\,n x^{3} + 66\,n x^{4} + 16\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-5 + 80\,x + 2\,n - 72\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x - 60\,x^{2} + 4\,n x + 64\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1	1							1
2	1							1
3	1	8						9
4	1	32	54					87
5	1	88	522	384				995
6	1	208	2952	7104	3000			13265
7	1	456	13068	70400	92700	25920		202545
8	1	960	50490	513920	1453500	1223424	246960	3489255

All 15 rows

Technical details

Node id: n-741874bf2346879a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3c9a530453f07c86

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1;
   3: 1,    8;
   4: 1,   32,     54;
   5: 1,   88,    522,      384;
   6: 1,  208,   2952,     7104,      3000;
   7: 1,  456,  13068,    70400,     92700,     25920;
   8: 1,  960,  50490,   513920,   1453500,   1223424,    246960;
   9: 1, 1976, 179550,  3132800,  16100500,  28270944,  16715664,   2580480;
  10: 1, 4016, 605340, 17000000, 144137500, 447842304, 540288504, 238970880, 29393280;

%F Formula

The row polynomials satisfy (2*n-5)*P(n,x) = -((16*n^2-72*n+72)*x-(72*n^3-512*n^2+1178*n-870)*x^2)*P(n-2,x) - ((4*n-6)*x-(16*n^2-56*n+50)*x^2-(32*n^3-280*n^2+740*n-600)*x^3)*d/dx P(n-2,x) - ((2*n-3)*x^2-(16*n^2-64*n+62)*x^3+(16*n^2-66*n+65)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-5+(16*n^2-72*n+80)*x)*P(n-1,x) + ((4*n-8)*x-(16*n^2-64*n+60)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((16*n^2 - 72*n + 72)*x - (72*n^3 - 512*n^2 + 1178*n - 870)*x^2)*P[n - 2, x] - ((4*n - 6)*x - (16*n^2 - 56*n + 50)*x^2 - (32*n^3 - 280*n^2 + 740*n - 600)*x^3)*D[P[n - 2, x], x] - ((2*n - 3)*x^2 - (16*n^2 - 64*n + 62)*x^3 + (16*n^2 - 66*n + 65)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 5 + (16*n^2 - 72*n + 80)*x)*P[n - 1, x] + ((4*n - 8)*x - (16*n^2 - 64*n + 60)*x^2)*D[P[n - 1, x], x])/(2*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-((16*n^2 - 72*n + 72)*x - (72*n^3 - 512*n^2 + 1178*n - 870)*x^2)*v[i - 2] - ((4*n - 6)*x - (16*n^2 - 56*n + 50)*x^2 - (32*n^3 - 280*n^2 + 740*n - 600)*x^3)*deriv(v[i - 2], x) - ((2*n - 3)*x^2 - (16*n^2 - 64*n + 62)*x^3 + (16*n^2 - 66*n + 65)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 5 + (16*n^2 - 72*n + 80)*x)*v[i - 1] + ((4*n - 8)*x - (16*n^2 - 64*n + 60)*x^2)*deriv(v[i - 1], x))/(2*n - 5)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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 of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A008517 → (θ+1)²

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.