A008955 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-n^{2} - 2\,n^{2} x - n^{2} x^{2} + 4\,n^{3} x + 4\,n^{3} x^{2} - 2\,n^{4} x - 6\,n^{4} x^{2} + 4\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 2\,n^{2} + 2\,n^{2} x - 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	5	8													13
3	14	98	108												220
4	30	546	2460	2304											5340
5	55	2046	22935	84304	72000										181340
6	91	6006	133419	1185184	3865680	3110400									8300780
7	140	14924	574860	9901892	76458200	230412384	177811200								495173600
8	204	32844	2007564	58956612	868609560	6102402144	17381935872	13005619200							37419564000
9	285	65892	5998110	275773524	6837966525	90531251376	594058938480	1622076254208	1185137049600						3498923298000
10	385	122892	15881910	1075521524	41309657025	911087234376	11156038265680	69514526366208	183668715648000	131681894400000					396974663098000
11	506	216062	38186808	3637803004	203982287530	6909249434406	139771186185092	1612235246534528	9646333617248064	24824920331520000	17526860144640000				53757237434056000
12	650	361790	84856200	10969670140	858786828250	42157388719590	1300525091165300	24614579030138240	270828443555841600	1568238299091210240	3949794240657408000	2753310393630720000			8568129508286920000
13	819	581490	176569965	30090600540	3176129645325	216319357488690	9612556900377795	275801710923779040	4950675281660874480	52423801589021466240	295485294041713391616	730951739489378304000	504085244567224320000		1588182388762577400000
14	1015	902538	347528025	76234218060	10548326777625	963345050069130	59077583312791575	2429014456608405120	65764952540354152800	1130570862928500797440	11598056916634741512960	63911080065499368947712	155709504596145217536000	106400762391727964160000	338818228851486027800000

Technical details

Node id: n-9729425cf37e269a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0ae9399177515f97

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   1;
  2:   5,     8;
  3:  14,    98,     108;
  4:  30,   546,    2460,      2304;
  5:  55,  2046,   22935,     84304,      72000;
  6:  91,  6006,  133419,   1185184,    3865680,     3110400;
  7: 140, 14924,  574860,   9901892,   76458200,   230412384,    177811200;
  8: 204, 32844, 2007564,  58956612,  868609560,  6102402144,  17381935872,   13005619200;
  9: 285, 65892, 5998110, 275773524, 6837966525, 90531251376, 594058938480, 1622076254208, 1185137049600;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-4*n^3+2*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*n + 1)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? 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.