A159854 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 12\,x + 16\,x^{2} + 2\,n x + 8\,n x^{2}\right)\,P_{n}(x) = \left(2 + 32\,x + 104\,x^{2} + 96\,x^{3} + 4\,n x + 24\,n x^{2} + 32\,n x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 16\,x + 48\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	16	48															65
1	2	40	192	256														490
2	4	96	624	1536	1280													3540
3	8	224	1824	6400	10240	6144												24840
4	16	512	4992	22528	52480	61440	28672											170640
5	32	1152	13056	71680	217600	374784	344064	131072										1153440
6	64	2560	33024	212992	793600	1794048	2437120	1835008	589824									7698240
7	128	5632	81408	602112	2652160	7397376	13246464	14811136	9437184	2621440								50855040
8	256	12288	196608	1638400	8314880	27525120	61014016	90177536	85524480	47185920	11534336							333123840
9	512	26624	466944	4325376	24821760	94961664	250478592	459276288	576847872	474480640	230686720	50331648						2166704640
10	1024	57344	1093632	11141120	71270400	309067776	944111616	2063597568	3220439040	3512729600	2549088256	1107296256	218103808					14007997440
11	2048	122880	2531328	28114944	198246400	960233472	3330539520	8443133952	15727067136	21338521600	20554186752	13337886720	5234491392	939524096				90094602240
12	4096	262144	5799936	69730304	537067520	2872049664	11142168576	32111591424	69448237056	112575119360	134997868544	116366770176	68266491904	24427626496	4026531840			576847319040
13	8192	557056	13172736	170393600	1422786560	8322023424	35687235584	115158810624	283398635520	533809070080	765326262272	821815148544	640788987904	342926295040	112742891520	17179869184		3678762147840
14	16384	1179648	29687808	411041792	3697541120	23473422336	110210580480	393459269632	1085011918848	2327167631360	3879412432896	4983235805184	4842776952832	3446174384128	1695169904640	515396075520	73014444032	23378642288640

Technical details

Node id: n-e4261e132ed380b5 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9ba80f07ba2d4a55

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,    16,     48;
  1:   2,    40,    192,     256;
  2:   4,    96,    624,    1536,     1280;
  3:   8,   224,   1824,    6400,    10240,     6144;
  4:  16,   512,   4992,   22528,    52480,    61440,     28672;
  5:  32,  1152,  13056,   71680,   217600,   374784,    344064,    131072;
  6:  64,  2560,  33024,  212992,   793600,  1794048,   2437120,   1835008,    589824;
  7: 128,  5632,  81408,  602112,  2652160,  7397376,  13246464,  14811136,   9437184,   2621440;
  8: 256, 12288, 196608, 1638400,  8314880, 27525120,  61014016,  90177536,  85524480,  47185920,  11534336;
  9: 512, 26624, 466944, 4325376, 24821760, 94961664, 250478592, 459276288, 576847872, 474480640, 230686720, 50331648;

%F Formula

The row polynomials satisfy (1+(2*n+12)*x+(8*n+16)*x^2)*P(n,x) = (2+(4*n+32)*x+(24*n+104)*x^2+(32*n+96)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 48*x^2.

%t Mathematica

P[0, x_] := 1 + 16*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (4*n + 32)*x + (24*n + 104)*x^2 + (32*n + 96)*x^3)*P[n - 1, x])/(1 + (2*n + 12)*x + (8*n + 16)*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 + 16*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (4*n + 32)*x + (24*n + 104)*x^2 + (32*n + 96)*x^3)*v[i - 1])/(1 + (2*n + 12)*x + (8*n + 16)*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_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.