A142963 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(24\,x + 60\,x^{2} + 32\,n x + 56\,n x^{2} - 32\,n^{2} x - 496\,n^{2} x^{2} + 288\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 16\,x^{2} + 80\,x^{3} - 4\,n x + 32\,n x^{2} + 128\,n x^{3} + 32\,n^{2} x^{2} - 640\,n^{2} x^{3} + 128\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 16\,x^{4} - 2\,n x^{2} + 32\,n x^{4} + 32\,n^{2} x^{3} - 128\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 8\,x + 2\,n - 32\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,x^{2} + 4\,n x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	8							9
2	1	40	36						77
3	1	120	648	128					897
4	1	296	4644	7424	400				12765
5	1	664	23256	112256	71600	1152			208929
6	1	1416	96660	1027840	2112000	625536	3136		3866589
7	1	2936	360072	7204480	33044000	34166016	5130496	8192	79916193

All 15 rows

Technical details

Node id: n-8601e37f53e74b20 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-23668f2803585b2c

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: 1,     8;
  2: 1,    40,      36;
  3: 1,   120,     648,       128;
  4: 1,   296,    4644,      7424,        400;
  5: 1,   664,   23256,    112256,      71600,        1152;
  6: 1,  1416,   96660,   1027840,    2112000,      625536,        3136;
  7: 1,  2936,  360072,   7204480,   33044000,    34166016,     5130496,       8192;
  8: 1,  5992, 1251972,  42900736,  367846000,   871163136,   500950912,   40271872,     20736;
  9: 1, 12120, 4160376, 229471872, 3313942800, 14761547136, 20107153920, 6865207296, 306001152, 51200;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = -((32*n^2-32*n-24)*x-(288*n^3-496*n^2+56*n+60)*x^2)*P(n-2,x) - ((4*n+2)*x-(32*n^2+32*n+16)*x^2-(128*n^3-640*n^2+128*n+80)*x^3)*d/dx P(n-2,x) - ((2*n+1)*x^2-32*n^2*x^3+(128*n^2-32*n-16)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+(32*n^2-32*n+8)*x)*P(n-1,x) + (4*n*x-(32*n^2-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*P[n - 2, x] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*D[P[n - 2, x], x] - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*P[n - 1, x] + (4*n*x - (32*n^2 - 8)*x^2)*D[P[n - 1, x], x])/(2*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] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*v[i - 2] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*deriv(v[i - 2], x) - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*v[i - 1] + (4*n*x - (32*n^2 - 8)*x^2)*deriv(v[i - 1], x))/(2*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, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.