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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	256	6144	49152	204800	519680	860160	953344	704512	334080	92160	11264							3735552
9	512	13312	116736	540672	1551360	2967552	3913728	3588096	2253312	926720	225280	24576						16121856
10	1024	28672	273408	1392640	4454400	9658368	14751744	16121856	12579840	6860800	2489344	540672	53248					69206016
11	2048	61440	632832	3514368	12390400	30007296	52039680	65961984	61433856	41676800	20072448	6512640	1277952	114688				295698432
12	4096	131072	1449984	8716288	33566720	89751552	174096384	250871808	271282176	219873280	131833856	56819712	16666624	2981888	245760			1258291200
13	8192	278528	3293184	21299200	88924160	260063232	557613056	899678208	1107025920	1042595840	747388928	401276928	156442624	41861120	6881280	524288		5335154688
14	16384	589824	7421952	51380224	231096320	733544448	1722040320	3073900544	4238327808	4545249280	3788488704	2433220608	1182318592	420675584	103464960	15728640	1114112	22548578304

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.