A112007 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-4\,x + 8\,n x + n^{2} - 7\,n^{2} x^{2} - 2\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(x - 5\,x^{2} + 10\,n x^{2} - 3\,n^{2} x + 4\,n^{2} x^{2} - 13\,n^{2} x^{3} - 2\,n^{3} x + 4\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} - x^{3} - 2\,n x^{2} + 2\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3} - 3\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-n + n^{2} + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-x + 2\,n x + 2\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	6	32	9						47
3	24	232	198	16					470
4	120	1776	2952	832	25				5705
5	720	14832	39600	23232	2850	36			81270
6	5040	135936	523260	513920	140250	8640	49		1327095
7	40320	1364544	7067736	10304320	4895000	718200	24206	64	24414390

All 15 rows

Technical details

Node id: n-6bbfb631a2c1156c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3d72d99fe3345b32

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:       2,         4;
  2:       6,        32,          9;
  3:      24,       232,        198,         16;
  4:     120,      1776,       2952,        832,         25;
  5:     720,     14832,      39600,      23232,       2850,         36;
  6:    5040,    135936,     523260,     513920,     140250,       8640,        49;
  7:   40320,   1364544,    7067736,   10304320,    4895000,     718200,     24206,       64;
  8:  362880,  14935680,   99236664,  199041024,  144137500,   38250000,   3295740,    64256,     81;
  9: 3628800, 177356160, 1459682208, 3822478464, 3883934600, 1611540000, 260981840, 14006272, 164106, 100;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -(n^4-n^2+(4*n^4-8*n+4)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - ((2*n^3+3*n^2-1)*x-(4*n^4+4*n^3+4*n^2+10*n-5)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n-1)*x^2-(4*n^3+4*n^2+2*n-1)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+n^2-n+4*n^3*x)*P(n-1,x) + ((2*n^2+2*n-1)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - n^2 + (4*n^4 - 8*n + 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - ((2*n^3 + 3*n^2 - 1)*x - (4*n^4 + 4*n^3 + 4*n^2 + 10*n - 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n - 1)*x^2 - (4*n^3 + 4*n^2 + 2*n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + n^2 - n + 4*n^3*x)*P[n - 1, x] + ((2*n^2 + 2*n - 1)*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - n^2 + (4*n^4 - 8*n + 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - ((2*n^3 + 3*n^2 - 1)*x - (4*n^4 + 4*n^3 + 4*n^2 + 10*n - 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n - 1)*x^2 - (4*n^3 + 4*n^2 + 2*n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + n^2 - n + 4*n^3*x)*v[i - 1] + ((2*n^2 + 2*n - 1)*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^2)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A163936 → (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.