A132393 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	12	42	36													90
3	60	282	432	216												990
4	360	2052	4284	3888	1296											11880
5	2520	16524	42300	52920	32400	7776										154440
6	20160	147312	437544	677160	576720	256608	46656									2162160
7	181440	1446768	4821768	8719704	9253440	5769792	1959552	279936								32432400
8	1814400	15556320	56898288	116127648	144852624	113218560	54214272	14556672	1679616							518918400
9	19958400	182005920	719219088	1618793856	2290144752	2114519904	1275668352	485409024	105815808	10077696						8821612800
10	239500800	2303821440	9722664576	23740840800	37194500160	39115107360	27995139648	13478918400	4182243840	755827200	60466176					158789030400
11	3113510400	31386683520	140217568128	366966917856	625973546880	731663396640	598627459584	343196777088	135242680320	34919216640	5321023488	362797056				3016991577600
12	43589145600	458094631680	2151366054912	5978842258752	10965431163456	13999128834240	12770764814016	8396519636736	3952578187008	1300325114880	284009628672	37005299712	2176782336			60339831552000
13	653837184000	7132954348800	35019058613760	102590830210752	200354521004352	275779519494336	275556245215680	202572383435136	109667790625536	43220345845248	12062095119360	2259137267712	254683533312	13060694016		1267136462592000
14	10461394944000	118050292684800	603102663912960	1851567635054592	3821217317334144	5614599437935488	6063577040416896	4894495606256256	2970118950619392	1349532277277184	452315596981248	108518766999552	17629760139264	1737072304128	78364164096	27877002177024000

Technical details

Node id: n-2d0bef3d53a72546 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-c93eb9c90ec44c16

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:        3,         6;
  2:       12,        42,        36;
  3:       60,       282,       432,        216;
  4:      360,      2052,      4284,       3888,       1296;
  5:     2520,     16524,     42300,      52920,      32400,       7776;
  6:    20160,    147312,    437544,     677160,     576720,     256608,      46656;
  7:   181440,   1446768,   4821768,    8719704,    9253440,    5769792,    1959552,    279936;
  8:  1814400,  15556320,  56898288,  116127648,  144852624,  113218560,   54214272,  14556672,   1679616;
  9: 19958400, 182005920, 719219088, 1618793856, 2290144752, 2114519904, 1275668352, 485409024, 105815808, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (n+2+6*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[(n + 2 + 6*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] = (n + 2 + 6*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= 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
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A028338 → basis A013611 → basis A013610

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.