A132166 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-3276 + 1260\,x + 108\,x^{2} + 2\,x^{3} - 720\,n + 180\,n x + 6\,n x^{2} - 36\,n^{2}\right)\,P_{n-2}(x) + \left(-1512\,x + 1182\,x^{2} + 70\,x^{3} + x^{4} - 216\,n x - 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(252 - 60\,x - 2\,x^{2} + 150\,n - 30\,n x - n x^{2} + 12\,n^{2}\right)\,P_{n-1}(x) + \left(72\,x + 2\,x^{2} + 36\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 378 + 44\,x + x^{2}\)

\(P_{1}(x) = 5649 + 1239\,x + 66\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	378	44	1								423
1	5649	1239	66	1							6955
2	77616	27846	2583	88	1						108134
3	1011024	542682	77175	4410	110	1					1635402
4	12690432	9567936	1937628	164220	6720	132	1				24367069
5	154664640	156444048	42945147	5036409	299565	9513	154	1			359399477
6	1835353728	2409645888	865268352	134197812	10844505	493794	12789	176	1		5255817045
7	21219988608	35323584192	16153121088	3208290228	337218336	20598921	757491	16548	198	1	76263575611

All 15 rows

Technical details

Node id: n-337b5a5e085dfa50 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-dcaa33841487c708

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           378,            44,             1;
  1:          5649,          1239,            66,             1;
  2:         77616,         27846,          2583,            88,            1;
  3:       1011024,        542682,         77175,          4410,          110,           1;
  4:      12690432,       9567936,       1937628,        164220,         6720,         132,          1;
  5:     154664640,     156444048,      42945147,       5036409,       299565,        9513,        154,        1;
  6:    1835353728,    2409645888,     865268352,     134197812,     10844505,      493794,      12789,      176,       1;
  7:   21219988608,   35323584192,   16153121088,    3208290228,    337218336,    20598921,     757491,    16548,     198,     1;
  8:  239049667584,  496287896832,  283090474368,   70290417600,   9319558950,   731792880,   35768187,  1101240,   20790,   220,   1;
  9: 2624349611520, 6715935338112, 4701583268928, 1432291307616, 234140023845, 22938119919, 1428657867, 58052358, 1535625, 25515, 242, 1;

%F Formula

The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+720*n+3276-(180*n+1260)*x-(6*n+108)*x^2-2*x^3)*P(n-2,x) - ((216*n+1512)*x+(6*n-1182)*x^2-70*x^3-x^4)*d/dx P(n-2,x) + (12*n^2+150*n+252-(30*n+60)*x-(n+2)*x^2)*P(n-1,x) + ((36*n+72)*x+(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 378 + 44*x + x^2, P(1,x) = 5649 + 1239*x + 66*x^2 + x^3.

%t Mathematica

P[0, x_] := 378 + 44*x + x^2;
P[1, x_] := 5649 + 1239*x + 66*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 720*n + 3276 - (180*n + 1260)*x - (6*n + 108)*x^2 - 2*x^3)*P[n - 2, x] - ((216*n + 1512)*x + (6*n - 1182)*x^2 - 70*x^3 - x^4)*D[P[n - 2, x], x] + (12*n^2 + 150*n + 252 - (30*n + 60)*x - (n + 2)*x^2)*P[n - 1, x] + ((36*n + 72)*x + (n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 378 + 44*x + x^2; v[2] = 5649 + 1239*x + 66*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 720*n + 3276 - (180*n + 1260)*x - (6*n + 108)*x^2 - 2*x^3)*v[i - 2] - ((216*n + 1512)*x + (6*n - 1182)*x^2 - 70*x^3 - x^4)*deriv(v[i - 2], x) + (12*n^2 + 150*n + 252 - (30*n + 60)*x - (n + 2)*x^2)*v[i - 1] + ((36*n + 72)*x + (n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 A convolution triangle of numbers obtained from A036224.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[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.