A257619 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-25 + 18\,x + 9\,n\right)\,P_{n}(x) = \left(352 - 2284\,x + 11116\,x^{2} - 12420\,x^{3} + 2592\,x^{4} - 198\,n + 1638\,n x - 12906\,n x^{2} + 10692\,n x^{3} - 1458\,n x^{4} - 324\,n^{2} x + 5346\,n^{2} x^{2} - 2187\,n^{2} x^{3} - 729\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(288\,x - 4212\,x^{2} + 9270\,x^{3} - 6804\,x^{4} + 1458\,x^{5} - 162\,n x + 3483\,n x^{2} - 5508\,n x^{3} + 2187\,n x^{4} - 729\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-307 + 1259\,x - 1044\,x^{2} + 162\,x^{3} + 117\,n - 819\,n x + 405\,n x^{2} + 162\,n^{2} x\right)\,P_{n-1}(x) + \left(-225\,x + 387\,x^{2} - 162\,x^{3} + 81\,n x - 81\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	48	52	4						104
3	572	1692	588	8					2860
4	6452	51556	41868	6500	16				106392
5	71436	1347092	2551376	923460	71564	32			4964960
6	787012	31727412	126455088	98875880	19404972	787332	64		278037760
7	8660140	701442404	5328558468	8303612120	3392678404	398351268	8660908	128	18141963840

All 15 rows

Technical details

Node id: n-939a2782d4f5adb9 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2f50bf37151e49a3

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:         48,           52,             4;
  3:        572,         1692,           588,              8;
  4:       6452,        51556,         41868,           6500,             16;
  5:      71436,      1347092,       2551376,         923460,          71564,             32;
  6:     787012,     31727412,     126455088,       98875880,       19404972,         787332,            64;
  7:    8660140,    701442404,    5328558468,     8303612120,     3392678404,      398351268,       8660908,          128;
  8:   95268708,  14920837636,  200736794116,   569300953404,   448421586764,   108918044060,    8079663756,     95270500,        256;
  9: 1047972428, 309944254356, 7006192228872, 33583551401436, 46949455058328, 21528397351692, 3365051684424, 162831916548, 1047976524, 512;

%F Formula

The row polynomials satisfy (9*n-25+18*x)*P(n,x) = -(198*n-352+(324*n^2-1638*n+2284)*x+(729*n^3-5346*n^2+12906*n-11116)*x^2+(2187*n^2-10692*n+12420)*x^3+(1458*n-2592)*x^4)*P(n-2,x) - ((162*n-288)*x+(729*n^2-3483*n+4212)*x^2-(729*n^2-5508*n+9270)*x^3-(2187*n-6804)*x^4-1458*x^5)*d/dx P(n-2,x) + (117*n-307+(162*n^2-819*n+1259)*x+(405*n-1044)*x^2+162*x^3)*P(n-1,x) + ((81*n-225)*x-(81*n-387)*x^2-162*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[(-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*P[n - 2, x] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*D[P[n - 2, x], x] + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*P[n - 1, x] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*D[P[n - 1, x], x])/(9*n - 25 + 18*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] = (-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*v[i - 2] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*deriv(v[i - 2], x) + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*v[i - 1] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*deriv(v[i - 1], x))/(9*n - 25 + 18*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) = 9*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) = 9*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.