A256890 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 4\,n + n^{2}\right)\,P_{n}(x) = \left(-12 - 52\,x + 100\,x^{2} - 10\,n - 92\,n x + 236\,n x^{2} - 2\,n^{2} - 48\,n^{2} x + 197\,n^{2} x^{2} - 8\,n^{3} x + 70\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-27\,x + 127\,x^{2} - 100\,x^{3} - 20\,n x + 149\,n x^{2} - 92\,n x^{3} - 4\,n^{2} x + 60\,n^{2} x^{2} - n^{2} x^{3} + 8\,n^{3} x^{2} + 18\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-7\,x^{2} + 51\,x^{3} - 44\,x^{4} - 5\,n x^{2} + 65\,n x^{3} - 60\,n x^{4} - n^{2} x^{2} + 28\,n^{2} x^{3} - 27\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(14 + 32\,x + 13\,n + 48\,n x + 3\,n^{2} + 24\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(11\,x - 48\,x^{2} + 9\,n x - 64\,n x^{2} + 2\,n^{2} x - 28\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 48\,x + 36\,x^{2}\)

\(P_{2}(x) = 8 + 208\,x + 468\,x^{2} + 128\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	4	48	36															88
2	8	208	468	128														812
3	16	784	3744	3136	400													8080
4	32	2736	23796	42304	17100	1152												87120
5	64	9104	132120	423040	367000	81936	3136											1016400
6	128	29360	671868	3524480	5507000	2687472	359660	8192										12788160
7	256	92688	3215952	25983424	66084000	58462704	17509072	1483008	20736									172851840
8	512	288304	14740740	175655104	680697700	980204688	537943756	104823040	5838156	51200								2500243200
9	1024	887440	65449800	1114127360	6279875000	13722865632	12308555000	4456509440	589048200	22186000	123904							38559628800
10	2048	2711472	283763340	6734188800	53346666000	168404219424	229216854216	136567464960	34091830800	3152926000	82022028	294912						631882944000
11	4096	8240912	1208263104	39220093760	425301696000	1870202327328	3667469667456	3324804137472	1377977495040	245125586000	16244426176	296672832	692224					10967859302400
12	8192	24952112	5074099092	221876748352	3225905537500	19215760713696	52249735849416	68244552946176	43235461605816	12903622150000	1677942909412	81185585472	1054226732	1605632				201062198937600
13	16384	75347856	21082387896	1226651809664	23515622256600	185608668709872	679545740953968	1228401953031168	1123330714638192	515579635305200	113815611721944	11039866286976	395880394936	3692044944	3686400			3882485198592000
14	32768	227092144	86872541724	6657976258176	166010098423400	1705610633563152	8215340161926772	19931317050964992	25225573142627568	16766000330462800	5732746851698372	956218166918784	70324874226984	1891890908656	12773933100	8388608		78777791059968000
15	65536	683504656	355665484080	35606482403520	1141703821188000	15046983477428112	93581028310278992	297414495785143040	504511462852551360	464710149664286000	231087845419260368	60187933909712448	7717917831230880	436179409443120	8891637102000	43744297984	18939904	1675880598712320000

Technical details

Node id: n-83d40d2edb3e8402 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5cf267279c5da1ec

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 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    4,      48,        36;
   2:    8,     208,       468,        128;
   3:   16,     784,      3744,       3136,         400;
   4:   32,    2736,     23796,      42304,       17100,         1152;
   5:   64,    9104,    132120,     423040,      367000,        81936,         3136;
   6:  128,   29360,    671868,    3524480,     5507000,      2687472,       359660,         8192;
   7:  256,   92688,   3215952,   25983424,    66084000,     58462704,     17509072,      1483008,       20736;
   8:  512,  288304,  14740740,  175655104,   680697700,    980204688,    537943756,    104823040,     5838156,      51200;
   9: 1024,  887440,  65449800, 1114127360,  6279875000,  13722865632,  12308555000,   4456509440,   589048200,   22186000,   123904;
  10: 2048, 2711472, 283763340, 6734188800, 53346666000, 168404219424, 229216854216, 136567464960, 34091830800, 3152926000, 82022028, 294912;

%F Formula

The row polynomials satisfy (n^2+4*n+4)*P(n,x) = -(2*n^2+10*n+12+(8*n^3+48*n^2+92*n+52)*x-(9*n^4+70*n^3+197*n^2+236*n+100)*x^2)*P(n-2,x) - ((4*n^2+20*n+27)*x-(8*n^3+60*n^2+149*n+127)*x^2-(4*n^4+18*n^3-n^2-92*n-100)*x^3)*d/dx P(n-2,x) - ((n^2+5*n+7)*x^2-(4*n^3+28*n^2+65*n+51)*x^3+(4*n^3+27*n^2+60*n+44)*x^4)*d^2/dx^2 P(n-2,x) + (3*n^2+13*n+14+(4*n^3+24*n^2+48*n+32)*x)*P(n-1,x) + ((2*n^2+9*n+11)*x-(4*n^3+28*n^2+64*n+48)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 4 + 48*x + 36*x^2, P(2,x) = 8 + 208*x + 468*x^2 + 128*x^3.

%t Mathematica

P[1, x_] := 4 + 48*x + 36*x^2;
P[2, x_] := 8 + 208*x + 468*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 10*n + 12 + (8*n^3 + 48*n^2 + 92*n + 52)*x - (9*n^4 + 70*n^3 + 197*n^2 + 236*n + 100)*x^2)*P[n - 2, x] - ((4*n^2 + 20*n + 27)*x - (8*n^3 + 60*n^2 + 149*n + 127)*x^2 - (4*n^4 + 18*n^3 - n^2 - 92*n - 100)*x^3)*D[P[n - 2, x], x] - ((n^2 + 5*n + 7)*x^2 - (4*n^3 + 28*n^2 + 65*n + 51)*x^3 + (4*n^3 + 27*n^2 + 60*n + 44)*x^4)*D[P[n - 2, x], {x, 2}] + (3*n^2 + 13*n + 14 + (4*n^3 + 24*n^2 + 48*n + 32)*x)*P[n - 1, x] + ((2*n^2 + 9*n + 11)*x - (4*n^3 + 28*n^2 + 64*n + 48)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 48*x + 36*x^2; v[2] = 8 + 208*x + 468*x^2 + 128*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 + 10*n + 12 + (8*n^3 + 48*n^2 + 92*n + 52)*x - (9*n^4 + 70*n^3 + 197*n^2 + 236*n + 100)*x^2)*v[i - 2] - ((4*n^2 + 20*n + 27)*x - (8*n^3 + 60*n^2 + 149*n + 127)*x^2 - (4*n^4 + 18*n^3 - n^2 - 92*n - 100)*x^3)*deriv(v[i - 2], x) - ((n^2 + 5*n + 7)*x^2 - (4*n^3 + 28*n^2 + 65*n + 51)*x^3 + (4*n^3 + 27*n^2 + 60*n + 44)*x^4)*deriv(deriv(v[i - 2], x), x) + (3*n^2 + 13*n + 14 + (4*n^3 + 24*n^2 + 48*n + 32)*x)*v[i - 1] + ((2*n^2 + 9*n + 11)*x - (4*n^3 + 28*n^2 + 64*n + 48)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n + 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.