A134991 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1 + 6\,x\)

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	6														7
3	1	20	45													66
4	1	50	315	420												786
5	1	112	1470	5040	4725											11348
6	1	238	5754	37800	86625	62370										192788
7	1	492	20475	227920	952875	1621620	945945									3769328
8	1	1002	68805	1211980	8183175	24594570	33108075	16216200								83383808
9	1	2024	222948	5948800	60610550	285044760	662161500	735134400	310134825							2059259808
10	1	4070	705276	27659632	407156750	2801258460	9955966020	18745927200	17677685025	6547290750						56163653184
11	1	8164	2195193	123803680	2554401850	24624960360	125250054930	354775861440	559793359125	458310352500	151242416325					1676677413568
12	1	16354	6757023	539118580	15248082850	199859559900	1393874552070	5557922370000	13020597114525	17644948571250	12754777110075	3794809718700				54382582971328
13	1	32736	20639034	2300624144	87696684075	1528515948960	14187342108940	76323368721600	248491290021975	494730748512000	586719747063450	379480971870000	102776096548125			1904328098775040
14	1	65502	62702766	9670314680	490247463615	11170580090670	134976766904980	951083160387360	4125424986057375	11297640452348250	19514808978414750	20555219309625000	12024803296130625	2988412653476250		71604040163981824
15	1	131036	189778599	40186125104	2680907291725	78789419670780	1218516580556275	11002366850995520	61738101649539315	222987737485012500	525348011202139125	800173577285082000	757562607656229375	404431845770452500	92854250304440625	2877398525487444480

Technical details

Node id: n-8fb868076fadaac9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5eb99b00596b7bfc

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,    6;
   3: 1,   20,     45;
   4: 1,   50,    315,      420;
   5: 1,  112,   1470,     5040,      4725;
   6: 1,  238,   5754,    37800,     86625,      62370;
   7: 1,  492,  20475,   227920,    952875,    1621620,     945945;
   8: 1, 1002,  68805,  1211980,   8183175,   24594570,   33108075,    16216200;
   9: 1, 2024, 222948,  5948800,  60610550,  285044760,  662161500,   735134400,   310134825;
  10: 1, 4070, 705276, 27659632, 407156750, 2801258460, 9955966020, 18745927200, 17677685025, 6547290750;

%F Formula

The row polynomials satisfy P(n,x) = -n^2*x^2*P(n-2,x) - (n*x^2+n*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-n^2*x^2*P[n - 2, x] - (n*x^2 + n*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 2)*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] = 1; v[2] = 1 + 6*x; for(n = 3, N, my(i = n); v[i] = -n^2*x^2*v[i - 2] - (n*x^2 + n*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 2)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

Related ?

Same fingerprint ?

A269939 → D

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.