A159854 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 4\,x + n + 4\,n x\right)\,P_{n}(x) = \left(8 + 32\,x + 32\,x^{2} + 2\,n + 12\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 8 + 32\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	8	32								40
1	20	128	192							340
2	48	416	1152	1024						2640
3	112	1216	4800	8192	5120					19440
4	256	3328	16896	41984	51200	24576				138240
5	576	8704	53760	174080	312320	294912	114688			959040
6	1280	22016	159744	634880	1495040	2088960	1605632	524288		6531840
7	2816	54272	451584	2121728	6164480	11354112	12959744	8388608	2359296	43856640

All 15 rows

Technical details

Node id: n-db2d48c86b57e427 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-657752d90232a0f4

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     8,     32;
  1:    20,    128,     192;
  2:    48,    416,    1152,     1024;
  3:   112,   1216,    4800,     8192,     5120;
  4:   256,   3328,   16896,    41984,    51200,     24576;
  5:   576,   8704,   53760,   174080,   312320,    294912,    114688;
  6:  1280,  22016,  159744,   634880,  1495040,   2088960,   1605632,    524288;
  7:  2816,  54272,  451584,  2121728,  6164480,  11354112,  12959744,   8388608,   2359296;
  8:  6144, 131072, 1228800,  6651904, 22937600,  52297728,  78905344,  76021760,  42467328,  10485760;
  9: 13312, 311296, 3244032, 19857408, 79134720, 214695936, 401866752, 512753664, 427032576, 209715200, 46137344;

%F Formula

The row polynomials satisfy (n+3+(4*n+4)*x)*P(n,x) = (2*n+8+(12*n+32)*x+(16*n+32)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 32*x.

%t Mathematica

P[0, x_] := 8 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*P[n - 1, x])/(n + 3 + (4*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*v[i - 1])/(n + 3 + (4*n + 4)*x)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.