A166961 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,x + n\right)\,P_{n}(x) = \left(18 - 81\,x + 196\,x^{2} - 196\,x^{3} + 48\,x^{4} - 6\,n + 31\,n x - 146\,n x^{2} + 156\,n x^{3} - 32\,n x^{4} - 4\,n^{2} x + 42\,n^{2} x^{2} - 24\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(27\,x - 210\,x^{2} + 340\,x^{3} - 168\,x^{4} + 32\,x^{5} - 9\,n x + 100\,n x^{2} - 104\,n x^{3} + 24\,n x^{4} - 14\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6\,x^{2} - 56\,x^{3} + 88\,x^{4} - 32\,x^{5} - 2\,n x^{2} + 28\,n x^{3} - 24\,n x^{4} - 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-27 + 88\,x - 80\,x^{2} + 16\,x^{3} + 7\,n - 31\,n x + 20\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-36\,x + 44\,x^{2} - 8\,x^{3} + 9\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(-8\,x^{2} + 8\,x^{3} + 2\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	10	11	1						22
3	60	143	45	1					249
4	362	2211	1387	158	1				4119
5	2176	34979	43213	10965	512	1			91846
6	13062	544271	1360755	635050	75435	1593	1		2630167
7	78380	8333871	41963281	34983245	7804060	471354	4859	1	93639051

All 15 rows

Technical details

Node id: n-521a49c8d2ed9a53 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3642e049ea0a05ff

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:       2,          1;
  2:      10,         11,           1;
  3:      60,        143,          45,           1;
  4:     362,       2211,        1387,         158,           1;
  5:    2176,      34979,       43213,       10965,         512,           1;
  6:   13062,     544271,     1360755,      635050,       75435,        1593,         1;
  7:   78380,    8333871,    41963281,    34983245,     7804060,      471354,      4859,        1;
  8:  470290,  126340523,  1258545715,  1859276286,   706874465,    84254374,   2755474,    14684,     1;
  9: 2821752, 1904983931, 36882647397, 95246084773, 59802056736, 12115556403, 824938786, 15378278, 44190, 1;

%F Formula

The row polynomials satisfy (n-4+4*x)*P(n,x) = -(6*n-18+(4*n^2-31*n+81)*x+(4*n^3-42*n^2+146*n-196)*x^2+(24*n^2-156*n+196)*x^3+(32*n-48)*x^4)*P(n-2,x) - ((9*n-27)*x+(14*n^2-100*n+210)*x^2-(4*n^2-104*n+340)*x^3-(24*n-168)*x^4-32*x^5)*d/dx P(n-2,x) - ((2*n-6)*x^2+(4*n^2-28*n+56)*x^3+(24*n-88)*x^4+32*x^5)*d^2/dx^2 P(n-2,x) + (7*n-27+(4*n^2-31*n+88)*x+(20*n-80)*x^2+16*x^3)*P(n-1,x) + ((9*n-36)*x-(2*n-44)*x^2-8*x^3)*d/dx P(n-1,x) + ((2*n-8)*x^2+8*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 18 + (4*n^2 - 31*n + 81)*x + (4*n^3 - 42*n^2 + 146*n - 196)*x^2 + (24*n^2 - 156*n + 196)*x^3 + (32*n - 48)*x^4)*P[n - 2, x] - ((9*n - 27)*x + (14*n^2 - 100*n + 210)*x^2 - (4*n^2 - 104*n + 340)*x^3 - (24*n - 168)*x^4 - 32*x^5)*D[P[n - 2, x], x] - ((2*n - 6)*x^2 + (4*n^2 - 28*n + 56)*x^3 + (24*n - 88)*x^4 + 32*x^5)*D[P[n - 2, x], {x, 2}] + (7*n - 27 + (4*n^2 - 31*n + 88)*x + (20*n - 80)*x^2 + 16*x^3)*P[n - 1, x] + ((9*n - 36)*x - (2*n - 44)*x^2 - 8*x^3)*D[P[n - 1, x], x] + ((2*n - 8)*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(n - 4 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 18 + (4*n^2 - 31*n + 81)*x + (4*n^3 - 42*n^2 + 146*n - 196)*x^2 + (24*n^2 - 156*n + 196)*x^3 + (32*n - 48)*x^4)*v[i - 2] - ((9*n - 27)*x + (14*n^2 - 100*n + 210)*x^2 - (4*n^2 - 104*n + 340)*x^3 - (24*n - 168)*x^4 - 32*x^5)*deriv(v[i - 2], x) - ((2*n - 6)*x^2 + (4*n^2 - 28*n + 56)*x^3 + (24*n - 88)*x^4 + 32*x^5)*deriv(deriv(v[i - 2], x), x) + (7*n - 27 + (4*n^2 - 31*n + 88)*x + (20*n - 80)*x^2 + 16*x^3)*v[i - 1] + ((9*n - 36)*x - (2*n - 44)*x^2 - 8*x^3)*deriv(v[i - 1], x) + ((2*n - 8)*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 4 + 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.