A111006 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4\,x + 14\,x^{2} - 10\,x^{3} - 10\,n x - 26\,n x^{2} + 50\,n x^{3} + n^{2} - 7\,n^{2} x - 50\,n^{2} x^{2} - 75\,n^{2} x^{3} + 25\,n^{2} x^{4} + 2\,n^{3} + 11\,n^{3} x + 20\,n^{3} x^{2} - 15\,n^{3} x^{3} - 50\,n^{3} x^{4} + n^{4} + 8\,n^{4} x + 26\,n^{4} x^{2} + 40\,n^{4} x^{3} + 25\,n^{4} x^{4}\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} - 18\,x^{3} - 26\,x^{4} - 10\,x^{5} + 12\,n x + 53\,n x^{2} + 79\,n x^{3} + 53\,n x^{4} + 15\,n x^{5} + 13\,n^{2} x + 87\,n^{2} x^{2} + 240\,n^{2} x^{3} + 286\,n^{2} x^{4} + 145\,n^{2} x^{5} + 25\,n^{2} x^{6} + 6\,n^{3} x + 49\,n^{3} x^{2} + 167\,n^{3} x^{3} + 269\,n^{3} x^{4} + 195\,n^{3} x^{5} + 50\,n^{3} x^{6} + n^{4} x + 9\,n^{4} x^{2} + 34\,n^{4} x^{3} + 66\,n^{4} x^{4} + 65\,n^{4} x^{5} + 25\,n^{4} x^{6}\right)\,P_{n-2}(x) + \left(2\,x - 35\,x^{2} - 45\,x^{3} + 5\,x^{4} + 25\,x^{5} + 4\,n x - 27\,n x^{2} - 122\,n x^{3} - 65\,n x^{4} + 5\,n^{2} x + 19\,n^{2} x^{2} + 5\,n^{2} x^{3} - 85\,n^{2} x^{4} - 50\,n^{2} x^{5} + 4\,n^{3} x + 27\,n^{3} x^{2} + 72\,n^{3} x^{3} + 65\,n^{3} x^{4} + n^{4} x + 8\,n^{4} x^{2} + 26\,n^{4} x^{3} + 40\,n^{4} x^{4} + 25\,n^{4} x^{5}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	4	18						22
3	0	0	18	48					66
4	0	0	9	80	125				214
5	0	0	0	48	250	288			586
6	0	0	0	16	225	720	637		1598
7	0	0	0	0	100	792	1862	1344	4098

All 15 rows

Technical details

Node id: n-ce9f6d305461a3cc · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-53fd623c7116840c

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: 0, 4;
  2: 0, 4, 18;
  3: 0, 0, 18, 48;
  4: 0, 0,  9, 80, 125;
  5: 0, 0,  0, 48, 250, 288;
  6: 0, 0,  0, 16, 225, 720,  637;
  7: 0, 0,  0,  0, 100, 792, 1862, 1344;
  8: 0, 0,  0,  0,  25, 504, 2499, 4544,  2754;
  9: 0, 0,  0,  0,   0, 180, 1960, 7104, 10530, 5500;

%F Formula

The row polynomials satisfy (n^4+2*n^3+n^2+(8*n^4+11*n^3-7*n^2-10*n-4)*x+(26*n^4+20*n^3-50*n^2-26*n+14)*x^2+(40*n^4-15*n^3-75*n^2+50*n-10)*x^3+(25*n^4-50*n^3+25*n^2)*x^4)*P(n,x) = ((n^4+6*n^3+13*n^2+12*n+4)*x+(9*n^4+49*n^3+87*n^2+53*n+2)*x^2+(34*n^4+167*n^3+240*n^2+79*n-18)*x^3+(66*n^4+269*n^3+286*n^2+53*n-26)*x^4+(65*n^4+195*n^3+145*n^2+15*n-10)*x^5+(25*n^4+50*n^3+25*n^2)*x^6)*P(n-2,x) + ((n^4+4*n^3+5*n^2+4*n+2)*x+(8*n^4+27*n^3+19*n^2-27*n-35)*x^2+(26*n^4+72*n^3+5*n^2-122*n-45)*x^3+(40*n^4+65*n^3-85*n^2-65*n+5)*x^4+(25*n^4-50*n^2+25)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x + (9*n^4 + 49*n^3 + 87*n^2 + 53*n + 2)*x^2 + (34*n^4 + 167*n^3 + 240*n^2 + 79*n - 18)*x^3 + (66*n^4 + 269*n^3 + 286*n^2 + 53*n - 26)*x^4 + (65*n^4 + 195*n^3 + 145*n^2 + 15*n - 10)*x^5 + (25*n^4 + 50*n^3 + 25*n^2)*x^6)*P[n - 2, x] + ((n^4 + 4*n^3 + 5*n^2 + 4*n + 2)*x + (8*n^4 + 27*n^3 + 19*n^2 - 27*n - 35)*x^2 + (26*n^4 + 72*n^3 + 5*n^2 - 122*n - 45)*x^3 + (40*n^4 + 65*n^3 - 85*n^2 - 65*n + 5)*x^4 + (25*n^4 - 50*n^2 + 25)*x^5)*P[n - 1, x])/(n^4 + 2*n^3 + n^2 + (8*n^4 + 11*n^3 - 7*n^2 - 10*n - 4)*x + (26*n^4 + 20*n^3 - 50*n^2 - 26*n + 14)*x^2 + (40*n^4 - 15*n^3 - 75*n^2 + 50*n - 10)*x^3 + (25*n^4 - 50*n^3 + 25*n^2)*x^4)]];
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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x + (9*n^4 + 49*n^3 + 87*n^2 + 53*n + 2)*x^2 + (34*n^4 + 167*n^3 + 240*n^2 + 79*n - 18)*x^3 + (66*n^4 + 269*n^3 + 286*n^2 + 53*n - 26)*x^4 + (65*n^4 + 195*n^3 + 145*n^2 + 15*n - 10)*x^5 + (25*n^4 + 50*n^3 + 25*n^2)*x^6)*v[i - 2] + ((n^4 + 4*n^3 + 5*n^2 + 4*n + 2)*x + (8*n^4 + 27*n^3 + 19*n^2 - 27*n - 35)*x^2 + (26*n^4 + 72*n^3 + 5*n^2 - 122*n - 45)*x^3 + (40*n^4 + 65*n^3 - 85*n^2 - 65*n + 5)*x^4 + (25*n^4 - 50*n^2 + 25)*x^5)*v[i - 1])/(n^4 + 2*n^3 + n^2 + (8*n^4 + 11*n^3 - 7*n^2 - 10*n - 4)*x + (26*n^4 + 20*n^3 - 50*n^2 - 26*n + 14)*x^2 + (40*n^4 - 15*n^3 - 75*n^2 + 50*n - 10)*x^3 + (25*n^4 - 50*n^3 + 25*n^2)*x^4)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Another version of Fibonacci-Pascal triangle A037027.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.