A269939 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	1	6													7
4	1	20	45												66
5	1	50	315	420											786
6	1	112	1470	5040	4725										11348
7	1	238	5754	37800	86625	62370									192788
8	1	492	20475	227920	952875	1621620	945945								3769328
9	1	1002	68805	1211980	8183175	24594570	33108075	16216200							83383808
10	1	2024	222948	5948800	60610550	285044760	662161500	735134400	310134825						2059259808
11	1	4070	705276	27659632	407156750	2801258460	9955966020	18745927200	17677685025	6547290750					56163653184
12	1	8164	2195193	123803680	2554401850	24624960360	125250054930	354775861440	559793359125	458310352500	151242416325				1676677413568
13	1	16354	6757023	539118580	15248082850	199859559900	1393874552070	5557922370000	13020597114525	17644948571250	12754777110075	3794809718700			54382582971328
14	1	32736	20639034	2300624144	87696684075	1528515948960	14187342108940	76323368721600	248491290021975	494730748512000	586719747063450	379480971870000	102776096548125		1904328098775040
15	1	65502	62702766	9670314680	490247463615	11170580090670	134976766904980	951083160387360	4125424986057375	11297640452348250	19514808978414750	20555219309625000	12024803296130625	2988412653476250	71604040163981824

Technical details

Node id: n-0490fcbf3cc940e2 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5eb99b00596b7bfc

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: 1;
   3: 1,    6;
   4: 1,   20,     45;
   5: 1,   50,    315,     420;
   6: 1,  112,   1470,    5040,     4725;
   7: 1,  238,   5754,   37800,    86625,     62370;
   8: 1,  492,  20475,  227920,   952875,   1621620,    945945;
   9: 1, 1002,  68805, 1211980,  8183175,  24594570,  33108075,  16216200;
  10: 1, 2024, 222948, 5948800, 60610550, 285044760, 662161500, 735134400, 310134825;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^2 - 2*n + 1)*x^2*v[i - 2] - ((n - 1)*x^2 + (n - 1)*x^3)*deriv(v[i - 2], x) + (1 + 2*n*x)*v[i - 1] + (x + x^2)*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, 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

Related ?

Same fingerprint ?

A134991 → θ+1

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.