A008556 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 4\,n x + 16\,n^{2} - 6\,n^{2} x + 16\,n^{3} + 6\,n^{3} x + n^{3} x^{2} + 12\,n^{4} x + 2\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-8 - 24\,x - 24\,x^{2} - 8\,x^{3} - 16\,n - 100\,n x - 36\,n x^{2} - 16\,n x^{3} + 32\,n^{2} - 36\,n^{2} x - 34\,n^{2} x^{2} - 2\,n^{2} x^{3} + 64\,n^{3} + 128\,n^{3} x + 52\,n^{3} x^{2} + 14\,n^{3} x^{3} + 48\,n^{4} x + 46\,n^{4} x^{2} + 10\,n^{4} x^{3} + 4\,n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 48\,x^{2} - 48\,x^{3} - 16\,x^{4} - 64\,n x - 152\,n x^{2} - 136\,n x^{3} - 48\,n x^{4} - 64\,n^{2} x - 184\,n^{2} x^{2} - 172\,n^{2} x^{3} - 52\,n^{2} x^{4} - 48\,n^{3} x^{2} - 72\,n^{3} x^{3} - 24\,n^{3} x^{4} - 4\,n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	6	8			14
3	20	48			68
4	70	240	54		364
5	252	1120	540		1912
6	924	5040	3780	320	10064
7	3432	22176	22680	4480	52768

All 15 rows

Technical details

Node id: n-4e47c1ce1e072796 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-baff7fa6b7300132

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:     6,      8;
  3:    20,     48;
  4:    70,    240,     54;
  5:   252,   1120,    540;
  6:   924,   5040,   3780,    320;
  7:  3432,  22176,  22680,   4480;
  8: 12870,  96096, 124740,  40320,  1750;
  9: 48620, 411840, 648648, 295680, 31500;

%F Formula

The row polynomials satisfy (16*n^3+16*n^2+4*n+(12*n^4+6*n^3-6*n^2-4*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (64*n^3+32*n^2-16*n-8+(48*n^4+128*n^3-36*n^2-100*n-24)*x+(4*n^5+46*n^4+52*n^3-34*n^2-36*n-24)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) - ((64*n^2+64*n+16)*x+(48*n^3+184*n^2+152*n+48)*x^2+(4*n^4+72*n^3+172*n^2+136*n+48)*x^3+(4*n^4+24*n^3+52*n^2+48*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[((64*n^3 + 32*n^2 - 16*n - 8 + (48*n^4 + 128*n^3 - 36*n^2 - 100*n - 24)*x + (4*n^5 + 46*n^4 + 52*n^3 - 34*n^2 - 36*n - 24)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] - ((64*n^2 + 64*n + 16)*x + (48*n^3 + 184*n^2 + 152*n + 48)*x^2 + (4*n^4 + 72*n^3 + 172*n^2 + 136*n + 48)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (12*n^4 + 6*n^3 - 6*n^2 - 4*n)*x + (n^5 + 2*n^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] = ((64*n^3 + 32*n^2 - 16*n - 8 + (48*n^4 + 128*n^3 - 36*n^2 - 100*n - 24)*x + (4*n^5 + 46*n^4 + 52*n^3 - 34*n^2 - 36*n - 24)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] - ((64*n^2 + 64*n + 16)*x + (48*n^3 + 184*n^2 + 152*n + 48)*x^2 + (4*n^4 + 72*n^3 + 172*n^2 + 136*n + 48)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (12*n^4 + 6*n^3 - 6*n^2 - 4*n)*x + (n^5 + 2*n^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 Triangle of coefficients of Legendre polynomials 2^n P_n (x).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) 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.