A137320 → reverse → basis A008277

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	2														2
2	6	2													8
3	24	24	4												52
4	120	264	116	12											512
5	720	3120	2680	680	48										7248
6	5040	41040	61160	28360	4696	240									140536
7	40320	604800	1469440	1104320	320432	37296	1440								3578048
8	362880	9959040	38030720	43415680	19586224	3884048	334512	10080							115583184
9	3628800	182165760	1068471040	1780378880	1169946848	350892192	50474080	3341376	80640						4609379616
10	39916800	3675974400	32625152000	77270636800	71010911776	30179484384	6438844928	701528544	36765504	725760					221979940896
11	479001600	81270604800	1080764608000	3571618265600	4465273780352	2581237149568	765391678656	121913789888	10397711808	441682560	7257600				12678795530432
12	6227020800	1955763532800	38720737932800	176182850905600	293808406401664	224642137754496	88859551259072	19391549555136	2391220581376	163843276608	5751192960	79833600			846128119246912
13	87178291200	50924576102400	1494857203840000	9273791400473600	20330315407770368	20149219070652672	10337627061334144	2966686471512192	495732858612992	48683526722688	2736779175936	80670021120	958003200		65150743162512512
14	1307674368000	1427021448652800	61951164846771200	520217879275494400	1482690420526771456	1876231625953228544	1223339164622560128	448857917777636224	97559134304604544	12870752689363712	1029718827905664	48323992624128	1212528821760	12454041600	5726225656922844160

Technical details

Node id: n-3d889b8995061d11 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d6e98125c0869a94

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): no

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,        24,          4;
  4:     120,       264,        116,         12;
  5:     720,      3120,       2680,        680,         48;
  6:    5040,     41040,      61160,      28360,       4696,       240;
  7:   40320,    604800,    1469440,    1104320,     320432,     37296,     1440;
  8:  362880,   9959040,   38030720,   43415680,   19586224,   3884048,   334512,   10080;
  9: 3628800, 182165760, 1068471040, 1780378880, 1169946848, 350892192, 50474080, 3341376, 80640;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx 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] + (n - 1)*x*D[P[n - 1, x], 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] + (n - 1)*x*deriv(v[i - 1], x)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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)

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.