A144445 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 7\,n\right)\,P_{n}(x) = \left(-49\,x^{2} - 343\,n x^{2}\right)\,P_{n-4}(x) + \left(-14\,x - 63\,x^{2} - 98\,n x + 1862\,n x^{2} - 686\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-49\,x^{2} + 49\,x^{3} - 343\,n x^{2} + 343\,n x^{3}\right)\,P_{n-3}'(x) + \left(-8 - 44\,x + 720\,x^{2} - 56\,n + 364\,n x - 1932\,n x^{2} - 98\,n^{2} x + 1568\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-7\,x - 245\,x^{2} + 252\,x^{3} - 49\,n x + 637\,n x^{2} - 588\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-47 + 156\,x + 63\,n - 266\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-42\,x + 42\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 23 + 2\,x\)

\(P_{3}(x) = 206 + 412\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	23	2						25
3	206	412	3					621
4	1677	12682	5031	4				19394
5	13452	266902	400353	53808	5			734520
6	107659	4807224	17748693	9614448	538295	6		32816325
7	861322	80048136	602387961	803183948	200120340	5167932	7	1691769646

All 15 rows

Technical details

Node id: n-a4556afe012af9bf · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c8a5291217c2fb84

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:        0;
  1:        1;
  2:       23,           2;
  3:      206,         412,            3;
  4:     1677,       12682,         5031,             4;
  5:    13452,      266902,       400353,         53808,             5;
  6:   107659,     4807224,     17748693,       9614448,        538295,            6;
  7:   861322,    80048136,    602387961,     803183948,     200120340,      5167932,           7;
  8:  6890633,  1276302958,  17625610338,   46750323452,   29376017230,   3828908874,    48234431,         8;
  9: 55125128, 19845666178, 470925473655, 2207410963968, 2759263704960, 941850947310, 69459831623, 441001024, 9;

%F Formula

The row polynomials satisfy (7*n-6)*P(n,x) = -(343*n+49)*x^2*P(n-4,x) - ((98*n+14)*x+(686*n^2-1862*n+63)*x^2)*P(n-3,x) - ((343*n+49)*x^2-(343*n+49)*x^3)*d/dx P(n-3,x) - (56*n+8+(98*n^2-364*n+44)*x+(343*n^3-1568*n^2+1932*n-720)*x^2)*P(n-2,x) - ((49*n+7)*x+(343*n^2-637*n+245)*x^2-(343*n^2-588*n+252)*x^3)*d/dx P(n-2,x) + (63*n-47+(98*n^2-266*n+156)*x)*P(n-1,x) + ((49*n-42)*x-(49*n-42)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 23 + 2*x, P(3,x) = 206 + 412*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 23 + 2*x;
P[3, x_] := 206 + 412*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(343*n + 49)*x^2*P[n - 4, x] - ((98*n + 14)*x + (686*n^2 - 1862*n + 63)*x^2)*P[n - 3, x] - ((343*n + 49)*x^2 - (343*n + 49)*x^3)*D[P[n - 3, x], x] - (56*n + 8 + (98*n^2 - 364*n + 44)*x + (343*n^3 - 1568*n^2 + 1932*n - 720)*x^2)*P[n - 2, x] - ((49*n + 7)*x + (343*n^2 - 637*n + 245)*x^2 - (343*n^2 - 588*n + 252)*x^3)*D[P[n - 2, x], x] + (63*n - 47 + (98*n^2 - 266*n + 156)*x)*P[n - 1, x] + ((49*n - 42)*x - (49*n - 42)*x^2)*D[P[n - 1, x], x])/(7*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 23 + 2*x; v[4] = 206 + 412*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(343*n + 49)*x^2*v[i - 4] - ((98*n + 14)*x + (686*n^2 - 1862*n + 63)*x^2)*v[i - 3] - ((343*n + 49)*x^2 - (343*n + 49)*x^3)*deriv(v[i - 3], x) - (56*n + 8 + (98*n^2 - 364*n + 44)*x + (343*n^3 - 1568*n^2 + 1932*n - 720)*x^2)*v[i - 2] - ((49*n + 7)*x + (343*n^2 - 637*n + 245)*x^2 - (343*n^2 - 588*n + 252)*x^3)*deriv(v[i - 2], x) + (63*n - 47 + (98*n^2 - 266*n + 156)*x)*v[i - 1] + ((49*n - 42)*x - (49*n - 42)*x^2)*deriv(v[i - 1], x))/(7*n - 6)); 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) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.