A390433 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 32\,x - x^{2} + 168\,n + 16\,n x - 73\,n^{2} - 2\,n^{2} x + 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(80\,x + 6\,x^{2} - 72\,n x - 2\,n x^{2} + 21\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-16\,x^{2} - x^{3} + 8\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(20 + 2\,x - 12\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-3\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	2														2
3	0	8	3													11
4	0	72	60	4												136
5	0	1152	1608	224	5											2989
6	0	28800	59616	13120	600	6										102142
7	0	1036800	2964384	919168	65000	1320	7									4986679
8	0	50803200	191275776	78405120	7648960	237720	2548	8								328373332
9	0	3251404800	15569542656	8095546368	1023530560	43410432	707952	4480	9							27984147257
10	0	263363788800	1561831372800	1000320500736	157507833600	8564599680	189404096	1817088	7344	10						2991779324154
11	0	26336378880000	189376641792000	146128593936384	27869224504320	1867670937600	52607954560	681636352	4166640	11400	11					391631803819267
12	0	3186701844480000	27309740986368000	24948234564304896	5645028745267200	453669030365184	15646585794560	257116282368	2116832256	8745000	16940	12				61559280998456416
13	0	458885065605120000	4620126279462912000	4926266962585583616	1301316760890501120	122913306267807744	5051143830053888	101187487823872	1053432262080	5850035840	17090040	24288	13			11434661765429214501
14	0	77551576087265280000	906233078373138432000	1114570234954373529600	339294924491200266240	37083525624465039360	1779969458249902080	42301418910193664	535208828633856	3750346096640	14706618784	31492032	33800	14		2476556149381515518070
15	0	15200108913103994880000	204018769998087045120000	286538290919483808153600	99449445971649843855360	12422216211716113072128	685833087656757196800	18954818373505925120	283616189701498368	2409843987033600	11905182043040	34182994560	55241368	45864	15	618333905958044177059823

Technical details

Node id: n-34db167902c8336b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c387372c4d7fcb32

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0,            2;
   3: 0,            8,             3;
   4: 0,           72,            60,             4;
   5: 0,         1152,          1608,           224,            5;
   6: 0,        28800,         59616,         13120,          600,          6;
   7: 0,      1036800,       2964384,        919168,        65000,       1320,         7;
   8: 0,     50803200,     191275776,      78405120,      7648960,     237720,      2548,       8;
   9: 0,   3251404800,   15569542656,    8095546368,   1023530560,   43410432,    707952,    4480,    9;
  10: 0, 263363788800, 1561831372800, 1000320500736, 157507833600, 8564599680, 189404096, 1817088, 7344, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4-14*n^3+73*n^2-168*n+144+(2*n^2-16*n+32)*x+x^2)*P(n-2,x) - ((2*n^3-21*n^2+72*n-80)*x+(2*n-6)*x^2)*d/dx P(n-2,x) - ((n^2-8*n+16)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2-12*n+20+2*x)*P(n-1,x) + (2*n-3)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*P[n - 2, x] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*D[P[n - 2, x], x] - ((n^2 - 8*n + 16)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 12*n + 20 + 2*x)*P[n - 1, x] + (2*n - 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*v[i - 2] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*deriv(v[i - 2], x) - ((n^2 - 8*n + 16)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 12*n + 20 + 2*x)*v[i - 1] + (2*n - 3)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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, Lah numbers with level 2.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.