A272867 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 24\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 24 + 32\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	24	32													56
2	288	1536	2304	1024											5152
3	2592	24192	82944	129024	92160	24576									355488
4	20736	276480	1492992	4202496	6635520	5898240	2752512	524288							21803264
5	155520	2695680	19906560	81838080	205701120	327352320	330301440	204472320	70778880	10485760					1253687680
6	1119744	23887872	223948800	1214300160	4220190720	9852420096	15755378688	17270046720	12740198400	6039797760	1660944384	201326592			69203559936
7	7838208	198567936	2257403904	15216574464	67722117120	209799217152	464932306944	745952772096	866843099136	721378344960	418557984768	160658620416	36641439744	3758096384	3713924383232

All 15 rows

Technical details

Node id: n-ba35e8a0ffd8c223 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-70c371a2e54a122c

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         0;
  1:        24,          32;
  2:       288,        1536,         2304,          1024;
  3:      2592,       24192,        82944,        129024,          92160,          24576;
  4:     20736,      276480,      1492992,       4202496,        6635520,        5898240,         2752512,          524288;
  5:    155520,     2695680,     19906560,      81838080,      205701120,      327352320,       330301440,       204472320,         70778880,         10485760;
  6:   1119744,    23887872,    223948800,    1214300160,     4220190720,     9852420096,     15755378688,     17270046720,      12740198400,       6039797760,       1660944384,        201326592;
  7:   7838208,   198567936,   2257403904,   15216574464,    67722117120,   209799217152,    464932306944,    745952772096,     866843099136,     721378344960,     418557984768,     160658620416,      36641439744,      3758096384;
  8:  53747712,  1576599552,  21069103104,  169890545664,   923027374080,  3574390063104,  10180947345408,  21703862255616,   34906105184256,   42363141488640,   38507334598656,   25773025001472,   12311523753984,   3968549781504,    773094113280,    68719476736;
  9: 362797056, 12093235200, 185752092672, 1744005758976, 11196723363840, 52106901848064, 181844419018752, 485725669687296, 1004877543112704, 1619085565624320, 2032039222050816, 1976202499719168, 1472106483154944, 823113302409216, 333976656936960, 92771293593600, 15771119910912, 1236950581248;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+24*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24 + 32*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 24 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 24*n*x + 16*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] = 24 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 24*n*x + 16*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_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.