A290596 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - x + n x\right)\,P_{n}(x) = \left(16 - 4\,x - 12\,n - 7\,n x + n x^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	2														4
3	10	20	3													33
4	80	240	72	4												396
5	880	3520	1584	176	5											6165
6	12320	61600	36960	6160	350	6										117396
7	209440	1256640	942480	209440	17850	612	7									2636469
8	4188800	29321600	26389440	7330400	833000	42840	980	8								68107068
9	96342400	770739200	809276160	269758720	38318000	2627520	90160	1472	9							1987153641
10	2504902400	22544121600	27052945920	10520590080	1793282400	153709920	7032480	172224	2106	10						64576759140
11	72642169600	726421696000	980669289600	435853017600	86675316000	8915175360	509854800	16648320	305370	2900	11					2311703475561
12	2324549427200	25570043699200	38355065548800	19177532774400	4358530176000	523023621120	35893777920	1465052160	35830080	510400	3872	12				90346140421164
13	81359229952000	976310759424000	1610912753049600	894951529472000	228822834240000	31381417267200	2512564454400	123064381440	3762158400	71456000	813120	5040	13			3826377986673213
14	3091650738176000	40191459596288000	72344627273318400	44210605555916800	12559831123840000	1937802516249600	177315262924800	10132300738560	371701249920	8824816000	133893760	1244880	6422	14		174523805028663156
15	126757680265216000	1774607523713024000	3460484671240396800	2306989780826931200	720934306508416000	123588738258585600	12722370114854400	830848660561920	35559419575680	1013088876800	19213754560	238187040	1843114	8036	15	8526952511550231165

Technical details

Node id: n-4c99c4c80965c995 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-53929619dc0d0749

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:          2,           2;
   3:         10,          20,           3;
   4:         80,         240,          72,           4;
   5:        880,        3520,        1584,         176,          5;
   6:      12320,       61600,       36960,        6160,        350,         6;
   7:     209440,     1256640,      942480,      209440,      17850,       612,       7;
   8:    4188800,    29321600,    26389440,     7330400,     833000,     42840,     980,      8;
   9:   96342400,   770739200,   809276160,   269758720,   38318000,   2627520,   90160,   1472,    9;
  10: 2504902400, 22544121600, 27052945920, 10520590080, 1793282400, 153709920, 7032480, 172224, 2106, 10;

%F Formula

The row polynomials satisfy -(4-(n-1)*x)*P(n,x) = -(12*n-16-(3*n^2-7*n-4)*x-n*x^2)*P(n-1,x) - (12*x-3*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 16 - (3*n^2 - 7*n - 4)*x - n*x^2)*P[n - 1, x] - (12*x - 3*n*x^2)*D[P[n - 1, x], x])/(-4 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(12*n - 16 - (3*n^2 - 7*n - 4)*x - n*x^2)*v[i - 1] - (12*x - 3*n*x^2)*deriv(v[i - 1], x))/(-4 + (n - 1)*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. A generalization of unsigned Lah numbers, called L[3,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.