A272867 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 - 4\,x + 8\,x^{2} + 12\,n + 16\,n x\right)\,P_{n}(x) = \left(18 + 144\,x + 384\,x^{2} + 384\,x^{3} + 128\,x^{4} + 72\,n + 384\,n x + 576\,n x^{2} + 256\,n x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	30	56	16													102
2	324	1824	3072	1792	256											7268
3	2808	26784	95040	156672	124416	43008	4096									452824
4	22032	297216	1631232	4700160	7686144	7225344	3735552	917504	65536							26280720
5	163296	2851200	21254400	88473600	226160640	368492544	384860160	251658240	96337920	18350080	1048576					1459650656
6	1166400	25007616	235892736	1288949760	4523765760	10696458240	17397448704	19520815104	14898954240	7455375360	2264924160	352321536	16777216			78677856832
7	8118144	206406144	2356687872	15969042432	71526260736	223343640576	499898843136	812371673088	960087195648	817694244864	490695819264	198709346304	50029658112	6576668672	268435456	4149742040448

All 15 rows

Technical details

Node id: n-e3c47c78f24a0470 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-86cc915b987fb8f3

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:        30,          56,           16;
  2:       324,        1824,         3072,          1792,            256;
  3:      2808,       26784,        95040,        156672,         124416,          43008,            4096;
  4:     22032,      297216,      1631232,       4700160,        7686144,        7225344,         3735552,          917504,            65536;
  5:    163296,     2851200,     21254400,      88473600,      226160640,      368492544,       384860160,       251658240,         96337920,         18350080,          1048576;
  6:   1166400,    25007616,    235892736,    1288949760,     4523765760,    10696458240,     17397448704,     19520815104,      14898954240,       7455375360,       2264924160,        352321536,         16777216;
  7:   8118144,   206406144,   2356687872,   15969042432,    71526260736,   223343640576,    499898843136,    812371673088,     960087195648,     817694244864,     490695819264,     198709346304,      50029658112,      6576668672,       268435456;
  8:  55427328,  1630347264,  21857402880,  176913580032,   965500010496,  3758995537920,  10776679022592,  23158283304960,   37619087966208,   46241597620224,   42743648747520,   29273691783168,   14459275837440,   4915590070272,   1056561954816,    120259084288,     4294967296;
  9: 372874752, 12456032256, 191798710272, 1805923123200, 11632724803584, 54346246520832, 190528902660096, 511703443832832, 1065593251823616, 1730738625970176, 2193947778613248, 2160933338087424, 1636790024798208, 936352262651904, 392770464251904, 115036404056064, 21569325760512, 2164663517184, 68719476736;

%F Formula

The row polynomials satisfy (12*n-9+(16*n-4)*x+8*x^2)*P(n,x) = (72*n+18+(384*n+144)*x+(576*n+384)*x^2+(256*n+384)*x^3+128*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((72*n + 18 + (384*n + 144)*x + (576*n + 384)*x^2 + (256*n + 384)*x^3 + 128*x^4)*P[n - 1, x])/(12*n - 9 + (16*n - 4)*x + 8*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((72*n + 18 + (384*n + 144)*x + (576*n + 384)*x^2 + (256*n + 384)*x^3 + 128*x^4)*v[i - 1])/(12*n - 9 + (16*n - 4)*x + 8*x^2)); 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.