A142459 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 + 8\,x + 4\,n\right)\,P_{n}(x) = \left(55 - 414\,x + 2615\,x^{2} - 2188\,x^{3} + 352\,x^{4} - 20\,n + 224\,n x - 2236\,n x^{2} + 1312\,n x^{3} - 128\,n x^{4} - 32\,n^{2} x + 656\,n^{2} x^{2} - 192\,n^{2} x^{3} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(44\,x - 736\,x^{2} + 1348\,x^{3} - 784\,x^{4} + 128\,x^{5} - 16\,n x + 432\,n x^{2} - 608\,n x^{3} + 192\,n x^{4} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-86 + 438\,x - 284\,x^{2} + 32\,x^{3} + 24\,n - 224\,n x + 80\,n x^{2} + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-60\,x + 92\,x^{2} - 32\,x^{3} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	11	12	1						24
4	60	177	62	1					300
5	309	2432	1986	312	1				5040
6	1558	28681	54248	19790	1562	1			105840
7	7807	304860	1206363	946720	187557	7812	1		2661120
8	39056	3040393	22827414	35544575	14612260	1734999	39062	1	77837760

All 15 rows

Technical details

Node id: n-84cea0d001714dc2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-84d0fce31cd67972

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      2,         1;
   3:     11,        12,          1;
   4:     60,       177,         62,           1;
   5:    309,      2432,       1986,         312,           1;
   6:   1558,     28681,      54248,       19790,        1562,           1;
   7:   7807,    304860,    1206363,      946720,      187557,        7812,          1;
   8:  39056,   3040393,   22827414,    35544575,    14612260,     1734999,      39062,         1;
   9: 195305,  29160096,  386272228,  1090742384,   861352830,   210824320,   15849524,    195312,      1;
  10: 976554, 272987313, 6055452896, 28819075948, 40268890164, 18550376726, 2925186832, 143817804, 976562, 1;

%F Formula

The row polynomials satisfy (4*n-15+8*x)*P(n,x) = -(20*n-55+(32*n^2-224*n+414)*x+(64*n^3-656*n^2+2236*n-2615)*x^2+(192*n^2-1312*n+2188)*x^3+(128*n-352)*x^4)*P(n-2,x) - ((16*n-44)*x+(64*n^2-432*n+736)*x^2-(64*n^2-608*n+1348)*x^3-(192*n-784)*x^4-128*x^5)*d/dx P(n-2,x) + (24*n-86+(32*n^2-224*n+438)*x+(80*n-284)*x^2+32*x^3)*P(n-1,x) + ((16*n-60)*x-(16*n-92)*x^2-32*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n - 55 + (32*n^2 - 224*n + 414)*x + (64*n^3 - 656*n^2 + 2236*n - 2615)*x^2 + (192*n^2 - 1312*n + 2188)*x^3 + (128*n - 352)*x^4)*P[n - 2, x] - ((16*n - 44)*x + (64*n^2 - 432*n + 736)*x^2 - (64*n^2 - 608*n + 1348)*x^3 - (192*n - 784)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (24*n - 86 + (32*n^2 - 224*n + 438)*x + (80*n - 284)*x^2 + 32*x^3)*P[n - 1, x] + ((16*n - 60)*x - (16*n - 92)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(4*n - 15 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(20*n - 55 + (32*n^2 - 224*n + 414)*x + (64*n^3 - 656*n^2 + 2236*n - 2615)*x^2 + (192*n^2 - 1312*n + 2188)*x^3 + (128*n - 352)*x^4)*v[i - 2] - ((16*n - 44)*x + (64*n^2 - 432*n + 736)*x^2 - (64*n^2 - 608*n + 1348)*x^3 - (192*n - 784)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (24*n - 86 + (32*n^2 - 224*n + 438)*x + (80*n - 284)*x^2 + 32*x^3)*v[i - 1] + ((16*n - 60)*x - (16*n - 92)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(4*n - 15 + 8*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).?
  - 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) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257618 → D+1

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.