A112493 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

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	2														3
3	1	8	9													18
4	1	22	75	60												158
5	1	52	390	840	525											1808
6	1	114	1638	7000	11025	5670										25448
7	1	240	6111	45472	133875	166320	72765									424784
8	1	494	21213	255220	1237775	2765070	2837835	1081080								8198688
9	1	1004	70308	1303720	9698150	34414380	61801740	54054000	18243225							179586528
10	1	2026	225984	6246064	68046550	357777420	994713720	1491890400	1137161025	344594250						4400657440
11	1	4072	711381	28600000	441731290	3289846560	13224100890	30124174080	38766853125	26189163000	7202019825					119267204224
12	1	8166	2207439	126730604	2709156450	27690242580	153979175350	497918781360	958916203245	1080302973750	655383804075	164991726900				3542021009920
13	1	16356	6781554	548127944	15921981075	218157259320	1627044038620	7150921858080	19273259780775	32112278698500	32164220538450	17709112020600	4111043861925			114382514963200
14	1	32738	20688138	2328142856	90572464255	1633751969850	15970610716060	92537473988960	334355079833775	770832181869750	1130923570426650	1019538877757400	513880482740625	110681950128750		3990446900759808
15	1	65504	62801019	9753918368	502335356965	11758877047008	148009110344095	1105342322564480	5195393541493155	15881445992412000	31942213475997825	41844101831532000	34292957548224375	15938200818540000	3201870700153125	149561806370449920

Technical details

Node id: n-d86098d7adb4744b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-ec8a74cefb032b84

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,    2;
   3: 1,    8,      9;
   4: 1,   22,     75,      60;
   5: 1,   52,    390,     840,      525;
   6: 1,  114,   1638,    7000,    11025,      5670;
   7: 1,  240,   6111,   45472,   133875,    166320,     72765;
   8: 1,  494,  21213,  255220,  1237775,   2765070,   2837835,    1081080;
   9: 1, 1004,  70308, 1303720,  9698150,  34414380,  61801740,   54054000,   18243225;
  10: 1, 2026, 225984, 6246064, 68046550, 357777420, 994713720, 1491890400, 1137161025, 344594250;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-4*n+4)*x^2*P(n-2,x) - ((n-2)*x^2+(n-2)*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 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4)*x^2*P[n - 2, x] - ((n - 2)*x^2 + (n - 2)*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 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 4*n + 4)*x^2*v[i - 2] - ((n - 2)*x^2 + (n - 2)*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 read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…?
  - 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.