A137320 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + x + n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	8	2														10
2	64	40	6													110
3	512	576	208	24												1320
4	4096	7168	4544	1232	120											17160
5	32768	81920	79360	37120	8352	720										240240
6	262144	884736	1208320	852480	326656	64224	5040									3603600
7	2097152	9175040	16744448	16486400	9433088	3127040	554112	40320								57657600
8	16777216	92274688	216530944	282591232	223842304	109914112	32576256	5309568	362880							980179200
9	134217728	905969664	2654994432	4426039296	4616650752	3117735936	1359751168	368239104	55998720	3628800						17643225600
10	1073741824	8724152320	31205621760	64613253120	85619638272	75725045760	45173104640	17903175680	4498619904	645016320	39916800					335221286400
11	8589934592	82678120448	354334801920	891373486080	1460316143616	1633236025344	1270085386240	685302661120	250827067392	59143569408	8059530240	479001600				6704425728000
12	68719476736	773094113280	3909493981184	11737340313600	23270384467968	32049998069760	31392751419392	21993531310080	10915551133696	3733900431360	833342644224	108605905920	6227020800			140792940288000
13	549755813888	7146825580544	42099269435392	148631638245376	350485840134144	582185367109632	699841984331776	615446770352128	395233847410688	182688919322624	58941347192832	12535644266496	1570298849280	87178291200		3097444686336000
14	4398046511104	65420941852672	443996539191296	1820542147493888	5033361294753792	9914770538889216	14331516381298688	15421203927793664	12393572334567424	7390019065741312	3211864567382016	984405362024448	200597054791680	24251909068800	1307674368000	71241227785728000

Technical details

Node id: n-fb9713b87484fc7d · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-027f711a8c8fc556

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         1;
  1:         8,         2;
  2:        64,        40,          6;
  3:       512,       576,        208,         24;
  4:      4096,      7168,       4544,       1232,        120;
  5:     32768,     81920,      79360,      37120,       8352,        720;
  6:    262144,    884736,    1208320,     852480,     326656,      64224,       5040;
  7:   2097152,   9175040,   16744448,   16486400,    9433088,    3127040,     554112,     40320;
  8:  16777216,  92274688,  216530944,  282591232,  223842304,  109914112,   32576256,   5309568,   362880;
  9: 134217728, 905969664, 2654994432, 4426039296, 4616650752, 3117735936, 1359751168, 368239104, 55998720, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (8+(n+1)*x)*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[(8 + (n + 1)*x)*P[n - 1, x]];
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] = (8 + (n + 1)*x)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    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
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A137320 → 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.