A257612 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-4 - 240\,x - 60\,x^{2} + 160\,n x + 64\,n x^{2} + 16\,n^{2} + 192\,n^{2} x + 608\,n^{2} x^{2} - 128\,n^{3} x - 1280\,n^{3} x^{2} + 576\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(64\,x - 240\,x^{2} - 80\,x^{3} + 32\,n x^{2} + 32\,n x^{3} - 64\,n^{2} x + 192\,n^{2} x^{2} + 896\,n^{2} x^{3} + 128\,n^{3} x^{2} - 1408\,n^{3} x^{3} + 256\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(48\,x^{2} - 32\,x^{3} - 16\,x^{4} - 64\,n^{2} x^{2} - 128\,n^{2} x^{3} + 192\,n^{2} x^{4} + 256\,n^{3} x^{3} - 256\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(4 - 8\,x - 8\,n + 48\,n x - 96\,n^{2} x + 64\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2} - 16\,n x + 16\,n x^{2} + 32\,n^{2} x + 32\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	4	96	36													136
3	8	736	1656	128												2528
4	16	4864	33120	19456	400											57856
5	32	30336	484416	861184	189600	1152										1566720
6	64	184832	6072768	24113152	16868800	1663488	3136									48906240
7	128	1115648	69876864	531912704	831113600	279507456	13666688	8192								1727201280
8	256	6709248	764034048	10179739648	29920089600	22904414208	4159740928	107347968	20736							68042096640
9	512	40290304	8093214720	177831706624	888490163200	1279425835008	544609601536	57551749120	815878656	51200						2956858490880
10	1024	241819648	84014355456	2921282011136	23217236019200	56294736740352	45505782597632	11685128044544	756129199104	6045491200	123904					140470596403200
11	2048	1451089920	860819134464	45976149196800	554845342617600	2107737524846592	2868864964374528	1420404077101056	232754255308800	9564657049600	43895470080	294912				7241053136486400
12	4096	8706916352	8745319342080	701819203616768	12429699093196800	70339544347705344	149180978147475456	125048078840365056	40272225061957632	4386370022604800	117575960043520	313448988672	692224			402485358152908800
13	8192	52242317312	88354359263232	10478452694843392	265405098892288000	2156028043252432896	6750903492698996736	8817506602708893696	4851063097317974016	1061620395569152000	79243298504753152	1413669748211712	2207237906432	1605632		23993752765228646400
14	16384	313455673344	889420853329920	153923760885334016	5464940238160281600	61955583962939523072	275299677478467059712	529050396162533621760	455087221954200649728	172098844341498675200	26450310752695762944	1385313847968006144	16701347134750720	15359327993856	3686400	1526963818940124364800

Technical details

Node id: n-70c952fab141f36d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5d3f47477444bdf9

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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,        8;
  2:   4,       96,         36;
  3:   8,      736,       1656,          128;
  4:  16,     4864,      33120,        19456,          400;
  5:  32,    30336,     484416,       861184,       189600,          1152;
  6:  64,   184832,    6072768,     24113152,     16868800,       1663488,         3136;
  7: 128,  1115648,   69876864,    531912704,    831113600,     279507456,     13666688,        8192;
  8: 256,  6709248,  764034048,  10179739648,  29920089600,   22904414208,   4159740928,   107347968,     20736;
  9: 512, 40290304, 8093214720, 177831706624, 888490163200, 1279425835008, 544609601536, 57551749120, 815878656, 51200;

%F Formula

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (16*n^2-4-(128*n^3-192*n^2-160*n+240)*x+(576*n^4-1280*n^3+608*n^2+64*n-60)*x^2)*P(n-2,x) - ((64*n^2-64)*x-(128*n^3+192*n^2+32*n-240)*x^2-(256*n^4-1408*n^3+896*n^2+32*n-80)*x^3)*d/dx P(n-2,x) - ((64*n^2-48)*x^2-(256*n^3-128*n^2-32)*x^3+(256*n^3-192*n^2+16)*x^4)*d^2/dx^2 P(n-2,x) - (8*n-4-(64*n^3-96*n^2+48*n-8)*x)*P(n-1,x) + ((32*n^2-16*n-8)*x-(64*n^3-32*n^2-16*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 4 - (128*n^3 - 192*n^2 - 160*n + 240)*x + (576*n^4 - 1280*n^3 + 608*n^2 + 64*n - 60)*x^2)*P[n - 2, x] - ((64*n^2 - 64)*x - (128*n^3 + 192*n^2 + 32*n - 240)*x^2 - (256*n^4 - 1408*n^3 + 896*n^2 + 32*n - 80)*x^3)*D[P[n - 2, x], x] - ((64*n^2 - 48)*x^2 - (256*n^3 - 128*n^2 - 32)*x^3 + (256*n^3 - 192*n^2 + 16)*x^4)*D[P[n - 2, x], {x, 2}] - (8*n - 4 - (64*n^3 - 96*n^2 + 48*n - 8)*x)*P[n - 1, x] + ((32*n^2 - 16*n - 8)*x - (64*n^3 - 32*n^2 - 16*n + 8)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 4 - (128*n^3 - 192*n^2 - 160*n + 240)*x + (576*n^4 - 1280*n^3 + 608*n^2 + 64*n - 60)*x^2)*v[i - 2] - ((64*n^2 - 64)*x - (128*n^3 + 192*n^2 + 32*n - 240)*x^2 - (256*n^4 - 1408*n^3 + 896*n^2 + 32*n - 80)*x^3)*deriv(v[i - 2], x) - ((64*n^2 - 48)*x^2 - (256*n^3 - 128*n^2 - 32)*x^3 + (256*n^3 - 192*n^2 + 16)*x^4)*deriv(deriv(v[i - 2], x), x) - (8*n - 4 - (64*n^3 - 96*n^2 + 48*n - 8)*x)*v[i - 1] + ((32*n^2 - 16*n - 8)*x - (64*n^3 - 32*n^2 - 16*n + 8)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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 Triangle read by rows: 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) = 4*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 read by rows: 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) = 4*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.