A257617 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-19 + 14\,x + 7\,n\right)\,P_{n}(x) = \left(216 - 1332\,x + 5172\,x^{2} - 5572\,x^{3} + 1176\,x^{4} - 126\,n + 966\,n x - 5866\,n x^{2} + 4900\,n x^{3} - 686\,n x^{4} - 196\,n^{2} x + 2450\,n^{2} x^{2} - 1029\,n^{2} x^{3} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(168\,x - 1960\,x^{2} + 4242\,x^{3} - 3136\,x^{4} + 686\,x^{5} - 98\,n x + 1617\,n x^{2} - 2548\,n x^{3} + 1029\,n x^{4} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-195 + 743\,x - 616\,x^{2} + 98\,x^{3} + 77\,n - 483\,n x + 245\,n x^{2} + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-133\,x + 231\,x^{2} - 98\,x^{3} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	40	44	4						88
3	396	1164	412	8					1980
4	3692	28508	23444	3740	16				59400
5	33596	599972	1132816	417460	33724	32			2217600
6	303324	11379628	44620336	35180120	7054484	303644	64		98841600
7	2732284	202395284	1497157620	2328876760	964933172	116224500	2733052	128	5115052800

All 15 rows

Technical details

Node id: n-35c6c8f05887aca3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0db3782eef894533

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:         4,           2;
  2:        40,          44,             4;
  3:       396,        1164,           412,             8;
  4:      3692,       28508,         23444,          3740,            16;
  5:     33596,      599972,       1132816,        417460,         33724,            32;
  6:    303324,    11379628,      44620336,      35180120,       7054484,        303644,           64;
  7:   2732284,   202395284,    1497157620,    2328876760,     964933172,     116224500,      2733052,         128;
  8:  24596188,  3459635708,   44937322780,  126192685444,   99979525940,   24711720292,   1889691412,    24597980,       256;
  9: 221378748, 57690799716, 1249502802472, 5892682174316, 8228639215608, 3812976052732, 608065672424, 30505736628, 221382844, 512;

%F Formula

The row polynomials satisfy (7*n-19+14*x)*P(n,x) = -(126*n-216+(196*n^2-966*n+1332)*x+(343*n^3-2450*n^2+5866*n-5172)*x^2+(1029*n^2-4900*n+5572)*x^3+(686*n-1176)*x^4)*P(n-2,x) - ((98*n-168)*x+(343*n^2-1617*n+1960)*x^2-(343*n^2-2548*n+4242)*x^3-(1029*n-3136)*x^4-686*x^5)*d/dx P(n-2,x) + (77*n-195+(98*n^2-483*n+743)*x+(245*n-616)*x^2+98*x^3)*P(n-1,x) + ((49*n-133)*x-(49*n-231)*x^2-98*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(126*n - 216 + (196*n^2 - 966*n + 1332)*x + (343*n^3 - 2450*n^2 + 5866*n - 5172)*x^2 + (1029*n^2 - 4900*n + 5572)*x^3 + (686*n - 1176)*x^4)*P[n - 2, x] - ((98*n - 168)*x + (343*n^2 - 1617*n + 1960)*x^2 - (343*n^2 - 2548*n + 4242)*x^3 - (1029*n - 3136)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (77*n - 195 + (98*n^2 - 483*n + 743)*x + (245*n - 616)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 133)*x - (49*n - 231)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 19 + 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(126*n - 216 + (196*n^2 - 966*n + 1332)*x + (343*n^3 - 2450*n^2 + 5866*n - 5172)*x^2 + (1029*n^2 - 4900*n + 5572)*x^3 + (686*n - 1176)*x^4)*v[i - 2] - ((98*n - 168)*x + (343*n^2 - 1617*n + 1960)*x^2 - (343*n^2 - 2548*n + 4242)*x^3 - (1029*n - 3136)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (77*n - 195 + (98*n^2 - 483*n + 743)*x + (245*n - 616)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 133)*x - (49*n - 231)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 19 + 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.?
    
    ↑ 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.