A075499 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-32\,x + 16\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-16\,x^{2} + 16\,x^{3}\right)\,P_{n-2}''(x) + \left(4 + 4\,x\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 16 + 48\,x + 9\,x^{2}\)

\(P_{1}(x) = 64 + 448\,x + 216\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	16	48	9								73
1	64	448	216	16							744
2	256	3840	3600	640	25						8361
3	1024	31744	51840	16640	1500	36					102784
4	4096	258048	693504	358400	56000	3024	49				1373121
5	16384	2080768	8902656	6967296	1680000	153216	5488	64			19805872
6	65536	16711680	111513600	127303680	44486400	6096384	362208	9216	81		306548785
7	262144	133955584	1375764480	2235105280	1088640000	210373632	18439680	768000	14580	100	5063323480

All 15 rows

Technical details

Node id: n-2cda667fec719db9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-c8bef37399d50d11

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 2)
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:      16,         48,            9;
  1:      64,        448,          216,           16;
  2:     256,       3840,         3600,          640,           25;
  3:    1024,      31744,        51840,        16640,         1500,           36;
  4:    4096,     258048,       693504,       358400,        56000,         3024,          49;
  5:   16384,    2080768,      8902656,      6967296,      1680000,       153216,        5488,         64;
  6:   65536,   16711680,    111513600,    127303680,     44486400,      6096384,      362208,       9216,        81;
  7:  262144,  133955584,   1375764480,   2235105280,   1088640000,    210373632,    18439680,     768000,     14580,     100;
  8: 1048576, 1072693248,  16810573824,  38207488000,  25265152000,   6616608768,   802652928,   48660480,   1496880,   22000,   121;
  9: 4194304, 8585740288, 204140445696, 641205272576, 565002240000, 195180429312, 31480221696, 2605498368, 115473600, 2728000, 31944, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(16*x-9*x^2)*P(n-2,x) - (32*x-16*x^2-4*x^3)*d/dx P(n-2,x) - (16*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (4+4*x)*P(n-1,x) + (8*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 16 + 48*x + 9*x^2, P(1,x) = 64 + 448*x + 216*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 16 + 48*x + 9*x^2;
P[1, x_] := 64 + 448*x + 216*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(16*x - 9*x^2)*P[n - 2, x] - (32*x - 16*x^2 - 4*x^3)*D[P[n - 2, x], x] - (16*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (4 + 4*x)*P[n - 1, x] + (8*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 16 + 48*x + 9*x^2; v[2] = 64 + 448*x + 216*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(16*x - 9*x^2)*v[i - 2] - (32*x - 16*x^2 - 4*x^3)*deriv(v[i - 2], x) - (16*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (4 + 4*x)*v[i - 1] + (8*x - 4*x^2)*deriv(v[i - 1], x)); 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 Stirling2 triangle with scaled diagonals (powers of 4).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.