A200536 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 21\,n x + 18\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 21 + 36\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	21	36													57
2	252	1314	2268	1296											5130
3	2268	19764	68607	118584	102060	34992									346275
4	18144	221616	1156680	3344004	5783400	5983632	3429216	839808							20776500
5	136080	2138400	14900760	60429240	157180905	271931580	312915960	230947200	99202320	18895680					1168678125
6	979776	18825696	164112480	856779120	2976375780	7224137766	12500778276	15422024160	13293110880	7624406880	2618941248	408146688			63108618750
7	6858432	155784384	1630592208	10413549216	45272080980	141482532492	327000714579	565930129968	733407711876	702914572080	484285885776	227133631872	64997360064	8571080448	3313202484375

All 15 rows

Technical details

Node id: n-d5de6dc9110cf4eb · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9a572e9b19125e69

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         0;
  1:        21,         36;
  2:       252,       1314,         2268,          1296;
  3:      2268,      19764,        68607,        118584,        102060,          34992;
  4:     18144,     221616,      1156680,       3344004,       5783400,        5983632,         3429216,          839808;
  5:    136080,    2138400,     14900760,      60429240,     157180905,      271931580,       312915960,       230947200,        99202320,         18895680;
  6:    979776,   18825696,    164112480,     856779120,    2976375780,     7224137766,     12500778276,     15422024160,     13293110880,       7624406880,       2618941248,        408146688;
  7:   6858432,  155784384,   1630592208,   10413549216,   45272080980,   141482532492,    327000714579,    565930129968,    733407711876,     702914572080,     484285885776,     227133631872,      64997360064,      8571080448;
  8:  47029248, 1232838144,  15061116672,  113746114944,  593905919040,  2270917464480,   6569182774224,  14639178709512,  25338276414864,   34063761967200,   35278011590976,   27640305931392,   15859355855616,   6291173048832,   1542794480640,    176319369216;
  9: 317447424, 9432723456, 131740680960, 1148570129664, 7001104531680, 31663079050176, 110019003684912, 299997477985968, 649909913990589, 1124990542447380, 1555983052115184, 1709806268709504, 1474432614371808, 976858895279232, 480194782099200, 165034929586176, 35407133330688, 3570467226624;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+21*n*x+18*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 21 + 36*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 21 + 36*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 21*n*x + 18*n*x^2)*P[n - 1, x])/(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] = 21 + 36*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 21*n*x + 18*n*x^2)*v[i - 1])/(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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.