A269939 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(8\,x + 12\,x^{2} - 8\,n x - 16\,n x^{2} + 2\,n^{2} x + 7\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 8\,x^{2} - 6\,x^{3} + n x + 6\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 6\,x - n - 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} + n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{3}(x) = 2\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	2														2
3	0	2	9													11
4	0	2	30	60												92
5	0	2	75	420	525											1022
6	0	2	168	1960	6300	5670										14100
7	0	2	357	7672	47250	103950	72765									231996
8	0	2	738	27300	284900	1143450	1891890	1081080								4429360
9	0	2	1503	91740	1514975	9819810	28693665	37837800	18243225							96202720
10	0	2	3036	297264	7436000	72732660	332552220	756756000	827026200	344594250						2341397632
11	0	2	6105	940368	34574540	488588100	3268134870	11378246880	21089168100	19641872250	7202019825					63103551040
12	0	2	12246	2926924	154754600	3065282220	28729120420	143142919920	399122844120	621992621250	504141387750	164991726900				1865343596352
13	0	2	24531	9009364	673898225	18297699420	233169486550	1592999488080	6252662666250	14467330127250	19409443428375	13914302301900	4111043861925			59999931991872
14	0	2	49104	27518712	2875780180	105236020890	1783268607120	16214105267360	85863789811800	276101433357750	544203823363200	640057905887400	411104386192500	110681950128750		2086118801984768
15	0	2	98253	83603688	12087893350	588296956338	13032343439115	154259162177120	1069968555435780	4583805540063750	12427404497583075	21288882521907000	22268154252093750	12949788165063750	3201870700153125	77957766206468096

Technical details

Node id: n-dac23666be2bdcf0 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-146205b457884abc

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: 0, 2;
   3: 0, 2,    9;
   4: 0, 2,   30,     60;
   5: 0, 2,   75,    420,     525;
   6: 0, 2,  168,   1960,    6300,     5670;
   7: 0, 2,  357,   7672,   47250,   103950,     72765;
   8: 0, 2,  738,  27300,  284900,  1143450,   1891890,   1081080;
   9: 0, 2, 1503,  91740, 1514975,  9819810,  28693665,  37837800,  18243225;
  10: 0, 2, 3036, 297264, 7436000, 72732660, 332552220, 756756000, 827026200, 344594250;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = ((2*n^2-8*n+8)*x-(n^3-7*n^2+16*n-12)*x^2)*P(n-2,x) + ((n-2)*x-(n^2-6*n+8)*x^2-(n^2-5*n+6)*x^3)*d/dx P(n-2,x) - (n-2-(2*n^2-8*n+6)*x)*P(n-1,x) + ((n-3)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 2*x + 9*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[3, x_] := 2*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*P[n - 2, x] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] - (n - 2 - (2*n^2 - 8*n + 6)*x)*P[n - 1, x] + ((n - 3)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 2*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = (((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*v[i - 2] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) - (n - 2 - (2*n^2 - 8*n + 6)*x)*v[i - 1] + ((n - 3)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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, 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.