A098158 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(24\,x - 72\,x^{2} + 40\,n x + 276\,n x^{2} - 2\,n^{2} - 102\,n^{2} x - 416\,n^{2} x^{2} - 3\,n^{3} + 30\,n^{3} x + 308\,n^{3} x^{2} + 20\,n^{4} x - 112\,n^{4} x^{2} + n^{5} - 8\,n^{5} x + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-8\,x - 8\,x^{2} + 16\,x^{3} - 16\,n x - 200\,n x^{2} - 120\,n x^{3} + 2\,n^{2} - 70\,n^{2} x^{2} + 3\,n^{3} - 39\,n^{3} x + 126\,n^{3} x^{2} + 252\,n^{3} x^{3} + 60\,n^{4} x^{2} - 208\,n^{4} x^{3} - n^{5} + 11\,n^{5} x - 40\,n^{5} x^{2} + 48\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x^{2} - 16\,x^{3} + 8\,x^{4} + 28\,n x^{2} + 28\,n x^{3} - 56\,n x^{4} + 2\,n^{2} x + 42\,n^{2} x^{2} - 52\,n^{2} x^{3} + 8\,n^{2} x^{4} + 4\,n^{3} x - 20\,n^{3} x^{2} - 48\,n^{3} x^{3} + 64\,n^{3} x^{4} + 2\,n^{4} x - 18\,n^{4} x^{2} + 48\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	4							4
3	0	4	9						13
4	0	0	27	16					43
5	0	0	9	96	25				130
6	0	0	0	80	250	36			366
7	0	0	0	16	375	540	49		980
8	0	0	0	0	175	1260	1029	64	2528

All 15 rows

Technical details

Node id: n-43db9268f0694442 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b05b95bae6fcd71d

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 -1)
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: 0, 4;
   3: 0, 4,  9;
   4: 0, 0, 27, 16;
   5: 0, 0,  9, 96,  25;
   6: 0, 0,  0, 80, 250,   36;
   7: 0, 0,  0, 16, 375,  540,   49;
   8: 0, 0,  0,  0, 175, 1260, 1029,   64;
   9: 0, 0,  0,  0,  25, 1008, 3430, 1792,   81;
  10: 0, 0,  0,  0,   0,  324, 4116, 8064, 2916, 100;

%F Formula

The row polynomials satisfy (n^5-3*n^3-2*n^2-(8*n^5-20*n^4-30*n^3+102*n^2-40*n-24)*x+(16*n^5-112*n^4+308*n^3-416*n^2+276*n-72)*x^2)*P(n,x) = -(n^5-3*n^3-2*n^2-(11*n^5-39*n^3-16*n-8)*x+(40*n^5-60*n^4-126*n^3+70*n^2+200*n+8)*x^2-(48*n^5-208*n^4+252*n^3-120*n+16)*x^3)*P(n-1,x) + ((2*n^4+4*n^3+2*n^2)*x-(18*n^4+20*n^3-42*n^2-28*n-8)*x^2+(48*n^4-48*n^3-52*n^2+28*n-16)*x^3-(32*n^4-64*n^3-8*n^2+56*n-8)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^3 - 2*n^2 - (11*n^5 - 39*n^3 - 16*n - 8)*x + (40*n^5 - 60*n^4 - 126*n^3 + 70*n^2 + 200*n + 8)*x^2 - (48*n^5 - 208*n^4 + 252*n^3 - 120*n + 16)*x^3)*P[n - 1, x] + ((2*n^4 + 4*n^3 + 2*n^2)*x - (18*n^4 + 20*n^3 - 42*n^2 - 28*n - 8)*x^2 + (48*n^4 - 48*n^3 - 52*n^2 + 28*n - 16)*x^3 - (32*n^4 - 64*n^3 - 8*n^2 + 56*n - 8)*x^4)*D[P[n - 1, x], x])/(n^5 - 3*n^3 - 2*n^2 - (8*n^5 - 20*n^4 - 30*n^3 + 102*n^2 - 40*n - 24)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = (-(n^5 - 3*n^3 - 2*n^2 - (11*n^5 - 39*n^3 - 16*n - 8)*x + (40*n^5 - 60*n^4 - 126*n^3 + 70*n^2 + 200*n + 8)*x^2 - (48*n^5 - 208*n^4 + 252*n^3 - 120*n + 16)*x^3)*v[i - 1] + ((2*n^4 + 4*n^3 + 2*n^2)*x - (18*n^4 + 20*n^3 - 42*n^2 - 28*n - 8)*x^2 + (48*n^4 - 48*n^3 - 52*n^2 + 28*n - 16)*x^3 - (32*n^4 - 64*n^3 - 8*n^2 + 56*n - 8)*x^4)*deriv(v[i - 1], x))/(n^5 - 3*n^3 - 2*n^2 - (8*n^5 - 20*n^4 - 30*n^3 + 102*n^2 - 40*n - 24)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).?
  - 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.