A088729 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-14 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-18 - 13\,x - 3\,x^{2}\right)\,P_{n-2}'(x) + \left(-6\,x - 2\,x^{2}\right)\,P_{n-2}''(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(6 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 9 + 2\,x\)

\(P_{2}(x) = 79 + 36\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	9	2															11
2	79	36	3														118
3	765	530	90	4													1389
4	8311	7740	1995	180	5												18231
5	100989	118202	40950	5600	315	6											266062
6	1362439	1920996	836703	153720	13090	504	7										4287459
7	20246445	33401090	17533530	4049724	463050	26964	756	8									75721567
8	328972471	621639180	381995715	106350300	15284535	1194480	50610	1080	9								1455488380
9	5805917469	12364629002	8706916350	2838239800	491673105	48284082	2744280	88440	1485	10							30258494023
10	110645911399	262204157556	208177133703	77783127840	15777801380	1867424328	133337589	5761800	146025	1980	11						676594803611
11	2265191981325	5911963380050	5224925675970	2201650895444	511949037600	70949681088	6104750652	331318416	11254815	230230	2574	12					16193078208176
12	49589790516631	141336865198620	137616580045035	64564884804420	16933623972265	2690338210560	271034764000	17723225520	755963208	20720700	349349	3276	13				413021617773597
13	1156201277261949	3573076648541402	3800350802470950	1964775996128400	573835012245495	102836788532478	11876688663840	909235218320	46720223550	1607585980	36306270	513240	4095	14			11183910813695983
14	28605950745797959	95275705230931716	109907104486315503	62082854933004360	19983575412852870	3988336685605272	519580418728371	45597945135600	2742798351150	113675201640	3222156938	60999120	733460	5040	15		320411565615819014
15	748605590542359405	2673206920351029410	3324284459672314410	2036997249793757164	716491494089976150	157615081091057772	22868330603105988	2263798447677624	156089328736770	7572153042740	258486119892	6140488536	98853300	1023400	6120	16	9682496850799548697

Technical details

Node id: n-e193497bf1792138 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5063296cf612be30

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            9,            2;
   2:           79,           36,            3;
   3:          765,          530,           90,           4;
   4:         8311,         7740,         1995,         180,           5;
   5:       100989,       118202,        40950,        5600,         315,          6;
   6:      1362439,      1920996,       836703,      153720,       13090,        504,         7;
   7:     20246445,     33401090,     17533530,     4049724,      463050,      26964,       756,       8;
   8:    328972471,    621639180,    381995715,   106350300,    15284535,    1194480,     50610,    1080,      9;
   9:   5805917469,  12364629002,   8706916350,  2838239800,   491673105,   48284082,   2744280,   88440,   1485,   10;
  10: 110645911399, 262204157556, 208177133703, 77783127840, 15777801380, 1867424328, 133337589, 5761800, 146025, 1980, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(14+6*x+x^2)*P(n-2,x) - (18+13*x+3*x^2)*d/dx P(n-2,x) - (6*x+2*x^2)*d^2/dx^2 P(n-2,x) + (9+2*x)*P(n-1,x) + (6+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 9 + 2*x, P(2,x) = 79 + 36*x + 3*x^2.

%t Mathematica

P[1, x_] := 9 + 2*x;
P[2, x_] := 79 + 36*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(14 + 6*x + x^2)*P[n - 2, x] - (18 + 13*x + 3*x^2)*D[P[n - 2, x], x] - (6*x + 2*x^2)*D[P[n - 2, x], {x, 2}] + (9 + 2*x)*P[n - 1, x] + (6 + 3*x)*D[P[n - 1, x], x] + 2*x*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] = 9 + 2*x; v[2] = 79 + 36*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(14 + 6*x + x^2)*v[i - 2] - (18 + 13*x + 3*x^2)*deriv(v[i - 2], x) - (6*x + 2*x^2)*deriv(deriv(v[i - 2], x), x) + (9 + 2*x)*v[i - 1] + (6 + 3*x)*deriv(v[i - 1], x) + 2*x*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.?
  - 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.