A159854 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 6\,x + 4\,x^{2} + n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(2 + 16\,x + 26\,x^{2} + 12\,x^{3} + 2\,n x + 6\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 8\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	8	12								21
1	2	20	48	32							102
2	4	48	156	192	80						480
3	8	112	456	800	640	192					2208
4	16	256	1248	2816	3280	1920	448				9984
5	32	576	3264	8960	13600	11712	5376	1024			44544
6	64	1280	8256	26624	49600	56064	38080	14336	2304		196608
7	128	2816	20352	75264	165760	231168	206976	115712	36864	5120	860160

All 15 rows

Technical details

Node id: n-6174136b207ae39c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c3f75468b073d346

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 2)
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,     8,     12;
  1:   2,    20,     48,     32;
  2:   4,    48,    156,    192,      80;
  3:   8,   112,    456,    800,     640,     192;
  4:  16,   256,   1248,   2816,    3280,    1920,     448;
  5:  32,   576,   3264,   8960,   13600,   11712,    5376,    1024;
  6:  64,  1280,   8256,  26624,   49600,   56064,   38080,   14336,    2304;
  7: 128,  2816,  20352,  75264,  165760,  231168,  206976,  115712,   36864,   5120;
  8: 256,  6144,  49152, 204800,  519680,  860160,  953344,  704512,  334080,  92160,  11264;
  9: 512, 13312, 116736, 540672, 1551360, 2967552, 3913728, 3588096, 2253312, 926720, 225280, 24576;

%F Formula

The row polynomials satisfy (1+(n+6)*x+(2*n+4)*x^2)*P(n,x) = (2+(2*n+16)*x+(6*n+26)*x^2+(4*n+12)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 12*x^2.

%t Mathematica

P[0, x_] := 1 + 8*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (2*n + 16)*x + (6*n + 26)*x^2 + (4*n + 12)*x^3)*P[n - 1, x])/(1 + (n + 6)*x + (2*n + 4)*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 + 8*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (2*n + 16)*x + (6*n + 26)*x^2 + (4*n + 12)*x^3)*v[i - 1])/(1 + (n + 6)*x + (2*n + 4)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 basis_map_A013609 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

Related ?

Same fingerprint ?

A391533 → reverse → θ+1

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.