A256890 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-6 - 6\,x - 6\,x^{2} - 2\,n - 8\,n x - 11\,n x^{2} - 2\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 3\,x^{2} + 6\,x^{3} - n x - 4\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 8\,x + 3\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 24\,x + 12\,x^{2}\)

\(P_{2}(x) = 8 + 104\,x + 156\,x^{2} + 32\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	4	24	12								40
2	8	104	156	32							300
3	16	392	1248	784	80						2520
4	32	1368	7932	10576	3420	192					23520
5	64	4552	44040	105760	73400	13656	448				241920
6	128	14680	223956	881120	1101400	447912	51380	1024			2721600
7	256	46344	1071984	6495856	13216800	9743784	2501296	185376	2304		33264000
8	512	144152	4913580	43913776	136139540	163367448	76849108	13102880	648684	5120	439084800

All 15 rows

Technical details

Node id: n-ca40f10a3c367093 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7c3b635378939853

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:    4,      24,       12;
   2:    8,     104,      156,         32;
   3:   16,     392,     1248,        784,          80;
   4:   32,    1368,     7932,      10576,        3420,         192;
   5:   64,    4552,    44040,     105760,       73400,       13656,         448;
   6:  128,   14680,   223956,     881120,     1101400,      447912,       51380,        1024;
   7:  256,   46344,  1071984,    6495856,    13216800,     9743784,     2501296,      185376,       2304;
   8:  512,  144152,  4913580,   43913776,   136139540,   163367448,    76849108,    13102880,     648684,      5120;
   9: 1024,  443720, 21816600,  278531840,  1255975000,  2287144272,  1758365000,   557063680,   65449800,   2218600,   11264;
  10: 2048, 1355736, 94587780, 1683547200, 10669333200, 28067369904, 32745264888, 17070933120, 3787981200, 315292600, 7456548, 24576;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(2*n+6+(2*n^2+8*n+6)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((n+3)*x+(n^2+4*n+3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (3*n+7+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 4 + 24*x + 12*x^2, P(2,x) = 8 + 104*x + 156*x^2 + 32*x^3.

%t Mathematica

P[1, x_] := 4 + 24*x + 12*x^2;
P[2, x_] := 8 + 104*x + 156*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 24*x + 12*x^2; v[2] = 8 + 104*x + 156*x^2 + 32*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.