A134991 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x - 2\,n - 4\,n x - n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x - x^{2} - n x - 2\,n x^{2} - n x^{3} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 3\,n + 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	3														3
3	10	30													40
4	25	210	315												550
5	56	980	3780	3780											8596
6	119	3836	28350	69300	51975										153580
7	246	13650	170940	762300	1351350	810810									3109296
8	501	45870	908985	6546540	20495475	28378350	14189175								70564896
9	1012	148632	4461600	48488440	237537300	567567000	643242600	275675400							1777121984
10	2035	470184	20744724	325725400	2334382050	8533685160	16402686300	15713497800	5892561675						49223755328
11	4082	1463462	92852760	2043521480	20520800300	107357189940	310428878760	497594097000	412479317250	137493105750					1488011230784
12	8177	4504682	404338935	12198466280	166549633250	1194749616060	4863182073750	11573864101800	15880453714125	11595251918250	3478575575475				48765233950784
13	16368	13759356	1725468108	70157347260	1273763290800	12160578950520	66782947631400	220881146686200	445257673660800	533381588239500	347857557547500	94870242967500			1722537395565312
14	32751	41801844	7252736010	392197970892	9308816742225	115694371632840	832197765338940	3667044432051000	10167876407113425	17740735434922500	18842284367156250	11099818427197500	2774954606799375		65250314121495552
15	65518	126519066	30139593828	2144725833380	65657849725650	1044442783333950	9627070994621080	54878312577368280	200688963736511250	477589101092853750	733492445844658500	699288560913442500	375543856786848750	86663966950811250	2638884554522186752

Technical details

Node id: n-a9776fec91eee7a3 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-064a056e3e3cfaf4

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    0;
   2:    3;
   3:   10,     30;
   4:   25,    210,      315;
   5:   56,    980,     3780,      3780;
   6:  119,   3836,    28350,     69300,      51975;
   7:  246,  13650,   170940,    762300,    1351350,     810810;
   8:  501,  45870,   908985,   6546540,   20495475,   28378350,    14189175;
   9: 1012, 148632,  4461600,  48488440,  237537300,  567567000,   643242600,   275675400;
  10: 2035, 470184, 20744724, 325725400, 2334382050, 8533685160, 16402686300, 15713497800, 5892561675;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2+4*n+2)*x+(n^3+2*n^2+n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2+2*n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+(2*n^2+4*n)*x)*P(n-1,x) + (n*x+n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 + 4*n + 2)*x + (n^3 + 2*n^2 + n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 + 2*n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 + 4*n)*x)*P[n - 1, x] + (n*x + n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 + 4*n + 2)*x + (n^3 + 2*n^2 + n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 + 2*n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + (2*n^2 + 4*n)*x)*v[i - 1] + (n*x + n*x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle of Ward numbers T(n,k) read by rows.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.