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	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

All 15 rows

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.