A115951 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 4\,n x + 12\,n^{2} + 6\,n^{2} x + 13\,n^{3} + 6\,n^{3} x + 4\,n^{3} x^{2} + 6\,n^{4} + n^{5}\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 16\,n + 100\,n x + 36\,n x^{2} + 16\,n x^{3} + 2\,n^{2} + 106\,n^{2} x + 12\,n^{2} x^{2} + 8\,n^{2} x^{3} - 14\,n^{3} + 48\,n^{3} x - 8\,n^{3} x^{2} - 10\,n^{4} + 10\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(16\,x + 48\,x^{2} + 48\,x^{3} + 16\,x^{4} + 48\,n x + 104\,n x^{2} + 88\,n x^{3} + 32\,n x^{4} + 52\,n^{2} x + 76\,n^{2} x^{2} + 40\,n^{2} x^{3} + 16\,n^{2} x^{4} + 24\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} x + 4\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	8														8
2	0	8	54													62
3	0	0	108	320												428
4	0	0	54	960	1750											2764
5	0	0	0	960	7000	9072										17032
6	0	0	0	320	10500	45360	45276									101456
7	0	0	0	0	7000	90720	271656	219648								589024
8	0	0	0	0	1750	90720	679140	1537536	1042470							3351616
9	0	0	0	0	0	45360	905520	4612608	8339760	4862000						18765248
10	0	0	0	0	0	9072	679140	7687680	29189160	43758000	22355476					103678528
11	0	0	0	0	0	0	271656	7687680	58378320	175032000	223554760	101582208				566506624
12	0	0	0	0	0	0	45276	4612608	72972900	408408000	1005996420	1117404288	457002364			3066441856
13	0	0	0	0	0	0	0	1537536	58378320	612612000	2682657120	5587021440	5484028368	2038517600		16464752384
14	0	0	0	0	0	0	0	219648	29189160	612612000	4694649960	16761064320	30162156024	26500728800	9026235000	87786854912

Technical details

Node id: n-8628db06a7bc9bde · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7b50549dba03cef1

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, 8;
  2: 0, 8,  54;
  3: 0, 0, 108, 320;
  4: 0, 0,  54, 960,  1750;
  5: 0, 0,   0, 960,  7000,  9072;
  6: 0, 0,   0, 320, 10500, 45360,  45276;
  7: 0, 0,   0,   0,  7000, 90720, 271656,  219648;
  8: 0, 0,   0,   0,  1750, 90720, 679140, 1537536, 1042470;
  9: 0, 0,   0,   0,     0, 45360, 905520, 4612608, 8339760, 4862000;

%F Formula

The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n+(6*n^3+6*n^2-4*n)*x+4*n^3*x^2)*P(n,x) = -(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8-(10*n^4+48*n^3+106*n^2+100*n+24)*x+(8*n^3-12*n^2-36*n-24)*x^2-(8*n^2+16*n+8)*x^3)*P(n-1,x) + ((4*n^4+24*n^3+52*n^2+48*n+16)*x+(4*n^4+24*n^3+76*n^2+104*n+48)*x^2+(40*n^2+88*n+48)*x^3+(16*n^2+32*n+16)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (10*n^4 + 48*n^3 + 106*n^2 + 100*n + 24)*x + (8*n^3 - 12*n^2 - 36*n - 24)*x^2 - (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (4*n^4 + 24*n^3 + 76*n^2 + 104*n + 48)*x^2 + (40*n^2 + 88*n + 48)*x^3 + (16*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n + (6*n^3 + 6*n^2 - 4*n)*x + 4*n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (10*n^4 + 48*n^3 + 106*n^2 + 100*n + 24)*x + (8*n^3 - 12*n^2 - 36*n - 24)*x^2 - (8*n^2 + 16*n + 8)*x^3)*v[i - 1] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (4*n^4 + 24*n^3 + 76*n^2 + 104*n + 48)*x^2 + (40*n^2 + 88*n + 48)*x^3 + (16*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n + (6*n^3 + 6*n^2 - 4*n)*x + 4*n^3*x^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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - 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.