A199478 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(16 + 8\,n + n^{2}\right)\,P_{n}(x) = \left(32 + 16\,x + 25\,n + 20\,n x + 11\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-32\,x - 32\,x^{2} - 36\,n - 41\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(27 + 51\,x + 24\,x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	4	3								7
2	8	9	2							19
3	16	23	8							47
4	32	57	30	4						123
5	64	135	92	20						311
6	128	313	266	88	8					803
7	256	711	720	312	48					2047
8	512	1593	1878	1020	240	16				5259
9	1024	3527	4740	3084	960	112				13447
10	2048	7737	11682	8880	3480	624	32			34483
11	4096	16839	28216	24528	11568	2768	256			88271
12	8192	36409	67054	65652	36288	10976	1568	64		226203
13	16384	78279	157164	171140	108480	39648	7616	576		579287
14	32768	167481	364090	436552	312360	134176	32704	3840	128	1484099

Technical details

Node id: n-6dacb02b1cf8b2b9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-0a329c2f7da3bd08

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half 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:    2,    1;
  1:    4,    3;
  2:    8,    9,    2;
  3:   16,   23,    8;
  4:   32,   57,   30,    4;
  5:   64,  135,   92,   20;
  6:  128,  313,  266,   88,   8;
  7:  256,  711,  720,  312,  48;
  8:  512, 1593, 1878, 1020, 240,  16;
  9: 1024, 3527, 4740, 3084, 960, 112;

%F Formula

The row polynomials satisfy (n^2+8*n+16)*P(n,x) = (11*n^2+25*n+32+(4*n^2+20*n+16)*x)*P(n-1,x) - (36*n+(41*n+32)*x+(8*n+32)*x^2)*d/dx P(n-1,x) + (27+51*x+24*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2 + x.

%t Mathematica

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n^2 + 25*n + 32 + (4*n^2 + 20*n + 16)*x)*P[n - 1, x] - (36*n + (41*n + 32)*x + (8*n + 32)*x^2)*D[P[n - 1, x], x] + (27 + 51*x + 24*x^2)*D[P[n - 1, x], {x, 2}])/(n^2 + 8*n + 16)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n^2 + 25*n + 32 + (4*n^2 + 20*n + 16)*x)*v[i - 1] - (36*n + (41*n + 32)*x + (8*n + 32)*x^2)*deriv(v[i - 1], x) + (27 + 51*x + 24*x^2)*deriv(deriv(v[i - 1], x), x))/(n^2 + 8*n + 16)); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Candidate equalities ?

A199478 → basis A007318 15 rows; no all-index equality certificate was found

Same fingerprint ?

A209599 → basis A013609

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003