A257609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(16\,x - 28\,n x^{2} - 16\,n^{2} x - 8\,n^{2} x^{2} + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 20\,x^{2} - 8\,n x + 32\,n x^{2} - 52\,n x^{3} + 16\,n^{2} x^{2} - 56\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} + 4\,x^{3} - 4\,n x^{2} + 16\,n x^{3} - 12\,n x^{4} + 16\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2\,n + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 4\,n x - 8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	8							10
2	4	64	36						104
3	8	352	792	128					1280
4	16	1664	9504	6656	400				18240
5	32	7296	86976	154624	45600	1152			295680
6	64	30720	686016	2473984	1905600	276480	3136		5376000
7	128	126464	4945536	31987712	49980800	19782144	1549184	8192	108380160

All 15 rows

Technical details

Node id: n-3c59e7c6cda68f8e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3b4269c91cbc7ccc

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,       8;
  2:   4,      64,        36;
  3:   8,     352,       792,        128;
  4:  16,    1664,      9504,       6656,         400;
  5:  32,    7296,     86976,     154624,       45600,        1152;
  6:  64,   30720,    686016,    2473984,     1905600,      276480,        3136;
  7: 128,  126464,   4945536,   31987712,    49980800,    19782144,     1549184,       8192;
  8: 256,  514048,  33656832,  361406464,   999616000,   813164544,   183242752,    8224768,    20736;
  9: 512, 2074624, 220446720, 3728932864, 16772531200, 24152444928, 11419856896, 1567621120, 42011136, 51200;

%F Formula

The row polynomials satisfy n*P(n,x) = -((16*n^2-16)*x-(36*n^3-8*n^2-28*n)*x^2)*P(n-2,x) - ((8*n+8)*x-(16*n^2+32*n+20)*x^2-(16*n^3-56*n^2-52*n)*x^3)*d/dx P(n-2,x) - ((4*n+4)*x^2-(16*n^2+16*n+4)*x^3+(16*n^2+12*n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+8*n^2*x)*P(n-1,x) + ((4*n+2)*x-(8*n^2+8*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((16*n^2 - 16)*x - (36*n^3 - 8*n^2 - 28*n)*x^2)*P[n - 2, x] - ((8*n + 8)*x - (16*n^2 + 32*n + 20)*x^2 - (16*n^3 - 56*n^2 - 52*n)*x^3)*D[P[n - 2, x], x] - ((4*n + 4)*x^2 - (16*n^2 + 16*n + 4)*x^3 + (16*n^2 + 12*n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + 8*n^2*x)*P[n - 1, x] + ((4*n + 2)*x - (8*n^2 + 8*n)*x^2)*D[P[n - 1, x], x])/(n)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((16*n^2 - 16)*x - (36*n^3 - 8*n^2 - 28*n)*x^2)*v[i - 2] - ((8*n + 8)*x - (16*n^2 + 32*n + 20)*x^2 - (16*n^3 - 56*n^2 - 52*n)*x^3)*deriv(v[i - 2], x) - ((4*n + 4)*x^2 - (16*n^2 + 16*n + 4)*x^3 + (16*n^2 + 12*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + 8*n^2*x)*v[i - 1] + ((4*n + 2)*x - (8*n^2 + 8*n)*x^2)*deriv(v[i - 1], x))/(n)); 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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

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.