A257627 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-18 + 14\,x + 7\,n\right)\,P_{n}(x) = \left(330 - 1839\,x + 5062\,x^{2} - 4991\,x^{3} + 1078\,x^{4} - 210\,n + 1365\,n x - 5432\,n x^{2} + 4606\,n x^{3} - 686\,n x^{4} - 294\,n^{2} x + 2303\,n^{2} x^{2} - 1029\,n^{2} x^{3} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(231\,x - 1911\,x^{2} + 3983\,x^{3} - 2989\,x^{4} + 686\,x^{5} - 147\,n x + 1568\,n x^{2} - 2450\,n x^{3} + 1029\,n x^{4} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-213 + 704\,x - 581\,x^{2} + 98\,x^{3} + 91\,n - 455\,n x + 245\,n x^{2} + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-126\,x + 224\,x^{2} - 98\,x^{3} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	3							9
2	69	78	9						156
3	780	2259	834	27					3900
4	8259	59382	50136	8502	81				126360
5	84534	1347303	2526024	968670	85506	243			5012280
6	852873	27549390	104285487	83706780	17609343	856518	729		234861120
7	8556432	527170371	3683214738	5706511125	2437842540	310633533	8569554	2187	12682500480

All 15 rows

Technical details

Node id: n-fdcf66a18d915ce2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2ea43ebad02ddf40

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 0)
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:         1;
  1:         6,            3;
  2:        69,           78,             9;
  3:       780,         2259,           834,             27;
  4:      8259,        59382,         50136,           8502,             81;
  5:     84534,      1347303,       2526024,         968670,          85506,            243;
  6:    852873,     27549390,     104285487,       83706780,       17609343,         856518,           729;
  7:   8556432,    527170371,    3683214738,     5706511125,     2437842540,      310633533,       8569554,        2187;
  8:  85662735,   9672012366,  116528057478,   320287911594,   256985536920,    66004028370,    5392771314,    85708662,      6561;
  9: 856968522, 172733246199, 3414654296808, 15545137443864, 21653348091732, 10266323644878, 1711139090136, 92793739392, 857125986, 19683;

%F Formula

The row polynomials satisfy (7*n-18+14*x)*P(n,x) = -(210*n-330+(294*n^2-1365*n+1839)*x+(343*n^3-2303*n^2+5432*n-5062)*x^2+(1029*n^2-4606*n+4991)*x^3+(686*n-1078)*x^4)*P(n-2,x) - ((147*n-231)*x+(343*n^2-1568*n+1911)*x^2-(343*n^2-2450*n+3983)*x^3-(1029*n-2989)*x^4-686*x^5)*d/dx P(n-2,x) + (91*n-213+(98*n^2-455*n+704)*x+(245*n-581)*x^2+98*x^3)*P(n-1,x) + ((49*n-126)*x-(49*n-224)*x^2-98*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(210*n - 330 + (294*n^2 - 1365*n + 1839)*x + (343*n^3 - 2303*n^2 + 5432*n - 5062)*x^2 + (1029*n^2 - 4606*n + 4991)*x^3 + (686*n - 1078)*x^4)*P[n - 2, x] - ((147*n - 231)*x + (343*n^2 - 1568*n + 1911)*x^2 - (343*n^2 - 2450*n + 3983)*x^3 - (1029*n - 2989)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (91*n - 213 + (98*n^2 - 455*n + 704)*x + (245*n - 581)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 126)*x - (49*n - 224)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 18 + 14*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(210*n - 330 + (294*n^2 - 1365*n + 1839)*x + (343*n^3 - 2303*n^2 + 5432*n - 5062)*x^2 + (1029*n^2 - 4606*n + 4991)*x^3 + (686*n - 1078)*x^4)*v[i - 2] - ((147*n - 231)*x + (343*n^2 - 1568*n + 1911)*x^2 - (343*n^2 - 2450*n + 3983)*x^3 - (1029*n - 2989)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (91*n - 213 + (98*n^2 - 455*n + 704)*x + (245*n - 581)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 126)*x - (49*n - 224)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 18 + 14*x)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.?
    
    ↑ Liu–Wang theorem (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.