A139548 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 12\,x^{2} + 8\,x^{3} + 2\,x^{4} + 12\,n x + 32\,n x^{2} + 28\,n x^{3} + 8\,n x^{4} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	2	16	18																											36
2	4	64	216	256	100																									640
3	8	192	1080	2560	3000	1728	392																							8960
4	16	512	4032	14336	28000	32256	21952	8192	1296																					110592
5	32	1280	12960	61440	168000	290304	329280	245760	116640	32000	3872																			1261568
6	64	3072	38016	225280	792000	1824768	2897664	3244032	2566080	1408000	511104	110592	10816																	13631488
7	128	7168	104832	745472	3203200	9225216	18834816	28114944	31135104	25625600	15503488	6709248	1968512	351232	28800															141557760
8	256	16384	276480	2293760	11648000	40255488	100452352	187432960	266872320	292864000	248055808	161021952	78740480	28098560	6912000	1048576	73984													1426063360
9	512	36864	705024	6684672	39168000	157925376	465733632	1042808832	1814731776	2489344000	2710895616	2346319872	1606305792	859815936	352512000	106954752	22639104	2985984	184832											14025752576
10	1024	81920	1751040	18677760	124032000	571539456	1944821760	5080350720	10448455680	17199104000	22892007424	24766709760	21799864320	15558574080	8930304000	4064280576	1433809920	378224640	70236160	8192000	451584									135291469824
11	2048	180224	4257792	50462720	374528000	1941553152	7487563776	22353543168	53046005760	101871616000	160244051968	208040361984	223811940352	199668367360	147350016000	89414172672	44161345536	17473978368	5408184320	1261568000	208631808	21807104	1083392							1284195221504
12	4096	393216	10174464	132644864	1088102400	6267469824	27013955584	90729086976	244011626496	535553638400	972029853696	1472285638656	1871881682944	2003944341504	1807493529600	1371017314304	870609383424	459316002816	199021182976	69638553600	19194126336	4012507136	598032384	56623104	2560000					12025908428800
13	8192	851968	23961600	340787200	3061760000	19399311360	92416163840	344876646400	1036650700800	2559631360000	5265161707520	9114149191680	13370583449600	16699536179200	17801072640000	16202931896320	12575468871680	8293205606400	4620134604800	2155479040000	831745474560	260812963840	64786841600	12268339200	1664000000	143982592	5971968			111325552312320
14	16384	1835008	55738368	858783744	8386560000	57967902720	302452899840	1241555927040	4124778577920	11316264960000	26016093143040	50663946977280	84234675732480	120236660490240	147885834240000	157043801456640	144046279802880	113993880698880	77618261360640	45265059840000	22457127813120	9389266698240	3265256816640	927486443520	209664000000	36283613184	4514807808	359661568	13778944	1020346790576128

Technical details

Node id: n-011c3414b63bdcbb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-561a7d876fc231a6

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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;
  1:   2,    16,     18;
  2:   4,    64,    216,     256,      100;
  3:   8,   192,   1080,    2560,     3000,      1728,       392;
  4:  16,   512,   4032,   14336,    28000,     32256,     21952,       8192,       1296;
  5:  32,  1280,  12960,   61440,   168000,    290304,    329280,     245760,     116640,      32000,       3872;
  6:  64,  3072,  38016,  225280,   792000,   1824768,   2897664,    3244032,    2566080,    1408000,     511104,     110592,      10816;
  7: 128,  7168, 104832,  745472,  3203200,   9225216,  18834816,   28114944,   31135104,   25625600,   15503488,    6709248,    1968512,    351232,     28800;
  8: 256, 16384, 276480, 2293760, 11648000,  40255488, 100452352,  187432960,  266872320,  292864000,  248055808,  161021952,   78740480,  28098560,   6912000,   1048576,    73984;
  9: 512, 36864, 705024, 6684672, 39168000, 157925376, 465733632, 1042808832, 1814731776, 2489344000, 2710895616, 2346319872, 1606305792, 859815936, 352512000, 106954752, 22639104, 2985984, 184832;

%F Formula

The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (2+(12*n+8)*x+(8*n^2+32*n+12)*x^2+(16*n^2+28*n+8)*x^3+(8*n^2+8*n+2)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (12*n + 8)*x + (8*n^2 + 32*n + 12)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (8*n^2 + 8*n + 2)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (12*n + 8)*x + (8*n^2 + 32*n + 12)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (8*n^2 + 8*n + 2)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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

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) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    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
  - ↑ Parent-row provers of polytool (above)
  - ↑ Iterated real-rootedness preservers (above)
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

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.