A137320 → reverse → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	2														2
2	6	2													8
3	24	20	4												48
4	120	172	80	12											384
5	720	1512	1168	392	48										3840
6	5040	14184	15736	8584	2296	240									46080
7	40320	143712	210992	163088	69872	15696	1440								645120
8	362880	1575648	2904912	2944736	1770464	630368	122832	10080							10321920
9	3628800	18659520	41654304	52686656	41262528	20467392	6271264	1083456	80640						185794560
10	39916800	237913920	626133024	954441952	928067712	596504064	253190432	68359392	10638144	725760					3715891200
11	479001600	3254135040	9892735488	17714633664	20681232064	16438725888	9003325824	3352217024	811251648	115090560	7257600				81749606400
12	6227020800	47572773120	164401046784	339110328000	463716987136	441196989248	297869220480	142615405376	47420658688	10419945408	1360344960	79833600			1961990553600
13	87178291200	740743073280	2872487932416	6720357153408	10561361755904	11741361695104	9464532957696	5571046321024	2375274086144	714927139968	144084174336	17441809920	958003200		51011754393600
14	1307674368000	12244463884800	52716978938880	138147700422528	245785069332864	313418128253312	294605696401792	206604623265408	108052713465472	41602470219392	11455315434624	2134721415168	241113576960	12454041600	1428329123020800

Technical details

Node id: n-2d6ac3c8b14b3d0e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-56ff70451877aec3

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:       2;
  2:       6,        2;
  3:      24,       20,        4;
  4:     120,      172,       80,       12;
  5:     720,     1512,     1168,      392,       48;
  6:    5040,    14184,    15736,     8584,     2296,      240;
  7:   40320,   143712,   210992,   163088,    69872,    15696,    1440;
  8:  362880,  1575648,  2904912,  2944736,  1770464,   630368,  122832,   10080;
  9: 3628800, 18659520, 41654304, 52686656, 41262528, 20467392, 6271264, 1083456, 80640;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(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[(n + 1 + (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] = (n + 1 + (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 (−∞, -1]
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), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n …?
  - Affine, reversal and Möbius substitutions (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
  - Affine, reversal and Möbius substitutions (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)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n …?
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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.