A257620 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(36\,x - 63\,n x^{2} - 36\,n^{2} x - 18\,n^{2} x^{2} + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 45\,x^{2} - 18\,n x + 72\,n x^{2} - 117\,n x^{3} + 36\,n^{2} x^{2} - 126\,n^{2} x^{3} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-9\,x^{2} + 9\,x^{3} - 9\,n x^{2} + 36\,n x^{3} - 27\,n x^{4} + 36\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(3\,n + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x + 6\,n x - 12\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	12							15
2	9	144	81						234
3	27	1188	2673	432					4320
4	81	8424	48114	33696	2025				92340
5	243	55404	660474	1174176	346275	8748			2245320
6	729	349920	7814151	28180224	21705975	3149280	35721		61236000
7	2187	2160756	84499119	546540048	853968825	337996476	26469261	139968	1851776640

All 15 rows

Technical details

Node id: n-890b628636ab287d · 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:     3,       12;
  2:     9,      144,         81;
  3:    27,     1188,       2673,          432;
  4:    81,     8424,      48114,        33696,         2025;
  5:   243,    55404,     660474,      1174176,       346275,         8748;
  6:   729,   349920,    7814151,     28180224,     21705975,      3149280,        35721;
  7:  2187,  2160756,   84499119,    546540048,    853968825,    337996476,     26469261,      139968;
  8:  6561, 13174488,  862587792,   9262452384,  25619064750,  20840517864,   4696311312,   210791808,     531441;
  9: 19683, 79755516, 8474712480, 143352706176, 644792444550, 928501120152, 439017662664, 60264622080, 1615049199, 1968300;

%F Formula

The row polynomials satisfy n*P(n,x) = -((36*n^2-36)*x-(81*n^3-18*n^2-63*n)*x^2)*P(n-2,x) - ((18*n+18)*x-(36*n^2+72*n+45)*x^2-(36*n^3-126*n^2-117*n)*x^3)*d/dx P(n-2,x) - ((9*n+9)*x^2-(36*n^2+36*n+9)*x^3+(36*n^2+27*n)*x^4)*d^2/dx^2 P(n-2,x) + (3*n+12*n^2*x)*P(n-1,x) + ((6*n+3)*x-(12*n^2+12*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((36*n^2 - 36)*x - (81*n^3 - 18*n^2 - 63*n)*x^2)*P[n - 2, x] - ((18*n + 18)*x - (36*n^2 + 72*n + 45)*x^2 - (36*n^3 - 126*n^2 - 117*n)*x^3)*D[P[n - 2, x], x] - ((9*n + 9)*x^2 - (36*n^2 + 36*n + 9)*x^3 + (36*n^2 + 27*n)*x^4)*D[P[n - 2, x], {x, 2}] + (3*n + 12*n^2*x)*P[n - 1, x] + ((6*n + 3)*x - (12*n^2 + 12*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] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-((36*n^2 - 36)*x - (81*n^3 - 18*n^2 - 63*n)*x^2)*v[i - 2] - ((18*n + 18)*x - (36*n^2 + 72*n + 45)*x^2 - (36*n^3 - 126*n^2 - 117*n)*x^3)*deriv(v[i - 2], x) - ((9*n + 9)*x^2 - (36*n^2 + 36*n + 9)*x^3 + (36*n^2 + 27*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (3*n + 12*n^2*x)*v[i - 1] + ((6*n + 3)*x - (12*n^2 + 12*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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ 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.