A257612 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(36 - 192\,x + 468\,x^{2} - 440\,x^{3} + 96\,x^{4} - 24\,n + 144\,n x - 488\,n x^{2} + 416\,n x^{3} - 64\,n x^{4} - 32\,n^{2} x + 208\,n^{2} x^{2} - 96\,n^{2} x^{3} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(24\,x - 176\,x^{2} + 360\,x^{3} - 272\,x^{4} + 64\,x^{5} - 16\,n x + 144\,n x^{2} - 224\,n x^{3} + 96\,n x^{4} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-36 + 112\,x - 92\,x^{2} + 16\,x^{3} + 16\,n - 72\,n x + 40\,n x^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 36\,x^{2} - 16\,x^{3} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	28	32	4						64
3	192	552	208	8					960
4	1232	8576	7328	1280	16				18432
5	7616	115232	215296	84160	7744	32			430080
6	46272	1395712	5195968	4206080	905792	46592	64		11796480
7	279040	15807104	107497728	166222720	72065024	9437568	279808	128	371589120

All 15 rows

Technical details

Node id: n-0c685534a1085bbf · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-04cae45d42f04794

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:       28,         32,           4;
  3:      192,        552,         208,            8;
  4:     1232,       8576,        7328,         1280,           16;
  5:     7616,     115232,      215296,        84160,         7744,           32;
  6:    46272,    1395712,     5195968,      4206080,       905792,        46592,          64;
  7:   279040,   15807104,   107497728,    166222720,     72065024,      9437568,      279808,       128;
  8:  1677568,  171462656,  1993593856,   5423448064,   4377972224,   1145589760,    96633856,   1679360,      256;
  9: 10073088, 1808564736, 34242691072, 153272907776, 213237639168, 102226496512, 17409204224, 979826688, 10077184, 512;

%F Formula

The row polynomials satisfy (2*n-5+4*x)*P(n,x) = -(24*n-36+(32*n^2-144*n+192)*x+(32*n^3-208*n^2+488*n-468)*x^2+(96*n^2-416*n+440)*x^3+(64*n-96)*x^4)*P(n-2,x) - ((16*n-24)*x+(32*n^2-144*n+176)*x^2-(32*n^2-224*n+360)*x^3-(96*n-272)*x^4-64*x^5)*d/dx P(n-2,x) + (16*n-36+(16*n^2-72*n+112)*x+(40*n-92)*x^2+16*x^3)*P(n-1,x) + ((8*n-20)*x-(8*n-36)*x^2-16*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[(-(24*n - 36 + (32*n^2 - 144*n + 192)*x + (32*n^3 - 208*n^2 + 488*n - 468)*x^2 + (96*n^2 - 416*n + 440)*x^3 + (64*n - 96)*x^4)*P[n - 2, x] - ((16*n - 24)*x + (32*n^2 - 144*n + 176)*x^2 - (32*n^2 - 224*n + 360)*x^3 - (96*n - 272)*x^4 - 64*x^5)*D[P[n - 2, x], x] + (16*n - 36 + (16*n^2 - 72*n + 112)*x + (40*n - 92)*x^2 + 16*x^3)*P[n - 1, x] + ((8*n - 20)*x - (8*n - 36)*x^2 - 16*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 4*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] = (-(24*n - 36 + (32*n^2 - 144*n + 192)*x + (32*n^3 - 208*n^2 + 488*n - 468)*x^2 + (96*n^2 - 416*n + 440)*x^3 + (64*n - 96)*x^4)*v[i - 2] - ((16*n - 24)*x + (32*n^2 - 144*n + 176)*x^2 - (32*n^2 - 224*n + 360)*x^3 - (96*n - 272)*x^4 - 64*x^5)*deriv(v[i - 2], x) + (16*n - 36 + (16*n^2 - 72*n + 112)*x + (40*n - 92)*x^2 + 16*x^3)*v[i - 1] + ((8*n - 20)*x - (8*n - 36)*x^2 - 16*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 4*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) = 4*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) = 4*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.