A269939 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 - 2\,x + n\right)\,P_{n}(x) = \left(8\,x^{2} + 12\,x^{3} + 4\,x^{4} - 12\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x^{2} - 14\,x^{3} - 10\,x^{4} - 2\,x^{5} + 5\,n x^{2} + 8\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3 + x + 6\,x^{2} + 2\,x^{3} + n - 7\,n x - 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 5\,x^{2} - 2\,x^{3} + n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	1	7	3													11
4	1	21	55	15												92
5	1	51	340	525	105											1022
6	1	113	1526	5530	5985	945										14100
7	1	239	5873	39718	96075	79695	10395									231996
8	1	493	20721	234745	1009855	1812195	1216215	135135								4429360
9	1	1003	69306	1234915	8486170	26231205	37207170	20945925	2027025							96202720
10	1	2025	223960	6023116	62097750	297166870	709668960	829728900	402026625	34459425						2341397632
11	1	4071	707311	27894724	414071658	2882689810	10422842430	20168208060	20020925925	8511477975	654729075					63103551040
12	1	8165	2199275	124535411	2585352770	25135840730	129354214990	372668726430	604140341805	520509614625	197073451575	13749310575				1865343596352
13	1	16355	6765200	541370921	15382862495	202909176470	1427184478720	5757047306010	13715337410775	19091681583975	14519271967200	4954334910525	316234143225			59999931991872
14	1	32737	20655402	2307503822	88271840111	1546055285775	14442094767100	78350131880020	258031711112175	522340891847775	636192821914650	432819130693950	134399510870625	7905853580625		2086118801984768
15	1	65503	62735517	9691215602	492665042285	11268629583393	136838530253425	970365555659500	4244310381105795	11756021006354625	20644573023649575	22329292853117250	13737738238599375	3913397522409375	213458046676875	77957766206468096

Technical details

Node id: n-47116cccf73aaef7 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-22d206e06a8a7627

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 -1)
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: 1;
   2: 1,    1;
   3: 1,    7,      3;
   4: 1,   21,     55,      15;
   5: 1,   51,    340,     525,      105;
   6: 1,  113,   1526,    5530,     5985,       945;
   7: 1,  239,   5873,   39718,    96075,     79695,     10395;
   8: 1,  493,  20721,  234745,  1009855,   1812195,   1216215,    135135;
   9: 1, 1003,  69306, 1234915,  8486170,  26231205,  37207170,  20945925,   2027025;
  10: 1, 2025, 223960, 6023116, 62097750, 297166870, 709668960, 829728900, 402026625, 34459425;

%F Formula

The row polynomials satisfy (n-3-2*x)*P(n,x) = -((n^3-6*n^2+12*n-8)*x^2-(3*n^2-12*n+12)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n^2-5*n+6)*x^2+(n^2-8*n+14)*x^3-(3*n-10)*x^4+2*x^5)*d/dx P(n-2,x) + (n-3+(2*n^2-7*n+1)*x-(5*n-6)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x+(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 - 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 - 2*x)); 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 read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
  - 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.