A257617 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 7\,n\right)\,P_{n}(x) = \left(20 - 240\,x - 120\,x^{2} + 70\,n - 700\,n x - 182\,n x^{2} + 490\,n^{2} x + 735\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(70\,x - 70\,x^{3} + 245\,n x - 98\,n x^{2} - 147\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-20 + 50\,x - 21\,n - 140\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-35\,x + 35\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	4	72	12						88
3	8	776	1164	32					1980
4	16	7352	37248	14704	80				59400
5	32	67128	849612	1132816	167820	192			2217600
6	64	606520	16614552	52109536	27690920	1819560	448		98841600
7	128	5464312	299494692	1863101408	2328876760	598989384	19125092	1024	5115052800

All 15 rows

Technical details

Node id: n-7076288e1d3dbb7d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-0fd6af0ce5e16283

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,         4;
  2:   4,        72,          12;
  3:   8,       776,        1164,            32;
  4:  16,      7352,       37248,         14704,            80;
  5:  32,     67128,      849612,       1132816,        167820,           192;
  6:  64,    606520,    16614552,      52109536,      27690920,       1819560,           448;
  7: 128,   5464312,   299494692,    1863101408,    2328876760,     598989384,      19125092,         1024;
  8: 256,  49191864,  5152559664,   57626403856,  139732605600,   86439605784,   12022639216,    196767456,       2304;
  9: 512, 442756472, 86204132220, 1627690788976, 6857199346340, 8228639215608, 2848458880708, 229877685920, 1992404124, 5120;

%F Formula

The row polynomials satisfy (7*n-5)*P(n,x) = (70*n+20+(490*n^2-700*n-240)*x-(343*n^3-735*n^2+182*n+120)*x^2)*P(n-2,x) + ((245*n+70)*x-(343*n^2+98*n)*x^2+(343*n^2-147*n-70)*x^3)*d/dx P(n-2,x) - (21*n+20-(98*n^2-140*n+50)*x)*P(n-1,x) + ((49*n-35)*x-(49*n-35)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((70*n + 20 + (490*n^2 - 700*n - 240)*x - (343*n^3 - 735*n^2 + 182*n + 120)*x^2)*P[n - 2, x] + ((245*n + 70)*x - (343*n^2 + 98*n)*x^2 + (343*n^2 - 147*n - 70)*x^3)*D[P[n - 2, x], x] - (21*n + 20 - (98*n^2 - 140*n + 50)*x)*P[n - 1, x] + ((49*n - 35)*x - (49*n - 35)*x^2)*D[P[n - 1, x], x])/(7*n - 5)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((70*n + 20 + (490*n^2 - 700*n - 240)*x - (343*n^3 - 735*n^2 + 182*n + 120)*x^2)*v[i - 2] + ((245*n + 70)*x - (343*n^2 + 98*n)*x^2 + (343*n^2 - 147*n - 70)*x^3)*deriv(v[i - 2], x) - (21*n + 20 - (98*n^2 - 140*n + 50)*x)*v[i - 1] + ((49*n - 35)*x - (49*n - 35)*x^2)*deriv(v[i - 1], x))/(7*n - 5)); 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) = 7*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) = 7*x + 2.?
    
    ↑ Liu–Wang theorem (above)

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.