A272866 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 20 + 32\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	20	32													52
2	200	1120	1920	1024											4264
3	1500	14400	52800	92160	76800	24576									262236
4	10000	136000	768000	2329600	4096000	4177920	2293760	524288							14335568
5	62500	1100000	8400000	36480000	99200000	175104000	200704000	144179200	58982400	10485760					734697860
6	375000	8100000	78000000	441600000	1632000000	4131840000	7311360000	9043968000	7667712000	4246732800	1384120320	201326592			36147134712
7	2187500	56000000	651000000	4547200000	21280000000	70425600000	169523200000	300482560000	392232960000	372506624000	250295091200	112742891520	30534533120	3758096384	1729037943724

All 15 rows

Technical details

Node id: n-90139221eebb3966 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-dc1f92b0a0065af1

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:       20,         32;
  2:      200,       1120,        1920,         1024;
  3:     1500,      14400,       52800,        92160,         76800,          24576;
  4:    10000,     136000,      768000,      2329600,       4096000,        4177920,        2293760,          524288;
  5:    62500,    1100000,     8400000,     36480000,      99200000,      175104000,      200704000,       144179200,        58982400,        10485760;
  6:   375000,    8100000,    78000000,    441600000,    1632000000,     4131840000,     7311360000,      9043968000,      7667712000,      4246732800,      1384120320,       201326592;
  7:  2187500,   56000000,   651000000,   4547200000,   21280000000,    70425600000,   169523200000,    300482560000,    392232960000,    372506624000,    250295091200,    112742891520,     30534533120,      3758096384;
  8: 12500000,  370000000,  5040000000,  41888000000,  237440000000,   971980800000,  2967552000000,   6879641600000,  12209356800000,  16588472320000,  17116954624000,  13176825446400,   7328287948800,   2780991324160,    644245094400,    68719476736;
  9: 70312500, 2362500000, 36900000000, 355680000000, 2368800000000, 11563776000000, 42835968000000, 122963558400000, 276885504000000, 491854233600000, 689291919360000, 757843623936000, 645805375488000, 417712413081600, 198105366528000, 64939905515520, 13142599925760, 1236950581248;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+20*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 20 + 32*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 20 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 20*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] = 20 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 20*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,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) 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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → D → basis A013611

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.