A261711 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1152\,n - 816\,n x + 392\,n x^{2} - 40\,n x^{3} - 1440\,n^{2} x + 624\,n^{2} x^{2} - 52\,n^{2} x^{3} + 48\,n^{3} x^{2} + 52\,n^{3} x^{3} + 2\,n^{3} x^{4} + 72\,n^{4} x^{3} + 4\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(576 - 1704\,x + 1660\,x^{2} - 516\,x^{3} - 12\,x^{4} - 4\,x^{5} + 4032\,n - 2784\,n x + 2752\,n x^{2} - 486\,n x^{3} - 82\,n x^{4} - 8\,n x^{5} - 4536\,n^{2} x - 108\,n^{2} x^{2} + 252\,n^{2} x^{3} - 55\,n^{2} x^{4} - n^{2} x^{5} - 48\,n^{3} x^{2} + 638\,n^{3} x^{3} + 99\,n^{3} x^{4} + 7\,n^{3} x^{5} + 240\,n^{4} x^{3} + 75\,n^{4} x^{4} + 5\,n^{4} x^{5} + 7\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(864 - 2160\,x + 1368\,x^{2} + 224\,x^{3} - 240\,x^{4} - 48\,x^{5} - 8\,x^{6} - 2592\,n x + 2520\,n x^{2} + 780\,n x^{3} - 488\,n x^{4} - 196\,n x^{5} - 24\,n x^{6} + 360\,n^{2} x^{2} + 348\,n^{2} x^{3} - 434\,n^{2} x^{4} - 248\,n^{2} x^{5} - 26\,n^{2} x^{6} + 216\,n^{3} x^{3} - 108\,n^{3} x^{4} - 96\,n^{3} x^{5} - 12\,n^{3} x^{6} + 6\,n^{4} x^{4} - 4\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	4				4
2	15	4			19
3	56	32			88
4	209	184	9		402
5	780	928	108		1816
6	2911	4364	837	16	8128
7	10864	19648	5328	256	36096

All 15 rows

Technical details

Node id: n-21e57a578117a548 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-a5ec587e1e490ed0

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:      4;
  2:     15,      4;
  3:     56,     32;
  4:    209,    184,      9;
  5:    780,    928,    108;
  6:   2911,   4364,    837,    16;
  7:  10864,  19648,   5328,   256;
  8:  40545,  85872,  30294,  2496,  25;
  9: 151316, 367296, 160056, 19200, 500;

%F Formula

The row polynomials satisfy (1152*n-(1440*n^2+816*n)*x+(48*n^3+624*n^2+392*n)*x^2+(72*n^4+52*n^3-52*n^2-40*n)*x^3+(2*n^5+4*n^4+2*n^3)*x^4)*P(n,x) = (4032*n+576-(4536*n^2+2784*n+1704)*x-(48*n^3+108*n^2-2752*n-1660)*x^2+(240*n^4+638*n^3+252*n^2-486*n-516)*x^3+(7*n^5+75*n^4+99*n^3-55*n^2-82*n-12)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) + (864-(2592*n+2160)*x+(360*n^2+2520*n+1368)*x^2+(216*n^3+348*n^2+780*n+224)*x^3+(6*n^4-108*n^3-434*n^2-488*n-240)*x^4-(4*n^4+96*n^3+248*n^2+196*n+48)*x^5-(2*n^4+12*n^3+26*n^2+24*n+8)*x^6)*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[((4032*n + 576 - (4536*n^2 + 2784*n + 1704)*x - (48*n^3 + 108*n^2 - 2752*n - 1660)*x^2 + (240*n^4 + 638*n^3 + 252*n^2 - 486*n - 516)*x^3 + (7*n^5 + 75*n^4 + 99*n^3 - 55*n^2 - 82*n - 12)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] + (864 - (2592*n + 2160)*x + (360*n^2 + 2520*n + 1368)*x^2 + (216*n^3 + 348*n^2 + 780*n + 224)*x^3 + (6*n^4 - 108*n^3 - 434*n^2 - 488*n - 240)*x^4 - (4*n^4 + 96*n^3 + 248*n^2 + 196*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(1152*n - (1440*n^2 + 816*n)*x + (48*n^3 + 624*n^2 + 392*n)*x^2 + (72*n^4 + 52*n^3 - 52*n^2 - 40*n)*x^3 + (2*n^5 + 4*n^4 + 2*n^3)*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; for(n = 1, N, my(i = n + 1); v[i] = ((4032*n + 576 - (4536*n^2 + 2784*n + 1704)*x - (48*n^3 + 108*n^2 - 2752*n - 1660)*x^2 + (240*n^4 + 638*n^3 + 252*n^2 - 486*n - 516)*x^3 + (7*n^5 + 75*n^4 + 99*n^3 - 55*n^2 - 82*n - 12)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] + (864 - (2592*n + 2160)*x + (360*n^2 + 2520*n + 1368)*x^2 + (216*n^3 + 348*n^2 + 780*n + 224)*x^3 + (6*n^4 - 108*n^3 - 434*n^2 - 488*n - 240)*x^4 - (4*n^4 + 96*n^3 + 248*n^2 + 196*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(1152*n - (1440*n^2 + 816*n)*x + (48*n^3 + 624*n^2 + 392*n)*x^2 + (72*n^4 + 52*n^3 - 52*n^2 - 40*n)*x^3 + (2*n^5 + 4*n^4 + 2*n^3)*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 Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string…?
  - 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.