A166962 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,x + n\right)\,P_{n}(x) = \left(24 - 119\,x + 404\,x^{2} - 474\,x^{3} + 120\,x^{4} - 8\,n + 47\,n x - 327\,n x^{2} + 360\,n x^{3} - 72\,n x^{4} - 6\,n^{2} x + 96\,n^{2} x^{2} - 54\,n^{2} x^{3} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(39\,x - 421\,x^{2} + 726\,x^{3} - 366\,x^{4} + 72\,x^{5} - 13\,n x + 213\,n x^{2} - 225\,n x^{3} + 54\,n x^{4} - 30\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(9\,x^{2} - 120\,x^{3} + 198\,x^{4} - 72\,x^{5} - 3\,n x^{2} + 63\,n x^{3} - 54\,n x^{4} - 9\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-35 + 128\,x - 122\,x^{2} + 24\,x^{3} + 9\,n - 47\,n x + 30\,n x^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-52\,x + 64\,x^{2} - 12\,x^{3} + 13\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x) + \left(-12\,x^{2} + 12\,x^{3} + 3\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	13	14	1						28
3	104	241	72	1					418
4	835	5174	3118	320	1				9448
5	6686	114505	137400	33146	1334	1			293072
6	53497	2491526	6175915	2777080	308255	5418	1		11811692
7	427988	53338489	272272836	221141895	47240260	2618603	21788	1	597061860

All 15 rows

Technical details

Node id: n-9de429b059cac5ba · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1f6d3ed798ab90b1

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:       13,          14,            1;
  3:      104,         241,           72,             1;
  4:      835,        5174,         3118,           320,            1;
  5:     6686,      114505,       137400,         33146,         1334,            1;
  6:    53497,     2491526,      6175915,       2777080,       308255,         5418,          1;
  7:   427988,    53338489,    272272836,     221141895,     47240260,      2618603,      21788,         1;
  8:  3423919,  1130807966,  11666247472,   17009865456,   6261487320,    708720534,   20923392,     87308,      1;
  9: 27391370, 23853108769, 488145521988, 1260684623920, 773443027716, 151011601338, 9672858564, 160346244, 349434, 1;

%F Formula

The row polynomials satisfy (n-4+4*x)*P(n,x) = -(8*n-24+(6*n^2-47*n+119)*x+(9*n^3-96*n^2+327*n-404)*x^2+(54*n^2-360*n+474)*x^3+(72*n-120)*x^4)*P(n-2,x) - ((13*n-39)*x+(30*n^2-213*n+421)*x^2-(9*n^2-225*n+726)*x^3-(54*n-366)*x^4-72*x^5)*d/dx P(n-2,x) - ((3*n-9)*x^2+(9*n^2-63*n+120)*x^3+(54*n-198)*x^4+72*x^5)*d^2/dx^2 P(n-2,x) + (9*n-35+(6*n^2-47*n+128)*x+(30*n-122)*x^2+24*x^3)*P(n-1,x) + ((13*n-52)*x-(3*n-64)*x^2-12*x^3)*d/dx P(n-1,x) + ((3*n-12)*x^2+12*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[(-(8*n - 24 + (6*n^2 - 47*n + 119)*x + (9*n^3 - 96*n^2 + 327*n - 404)*x^2 + (54*n^2 - 360*n + 474)*x^3 + (72*n - 120)*x^4)*P[n - 2, x] - ((13*n - 39)*x + (30*n^2 - 213*n + 421)*x^2 - (9*n^2 - 225*n + 726)*x^3 - (54*n - 366)*x^4 - 72*x^5)*D[P[n - 2, x], x] - ((3*n - 9)*x^2 + (9*n^2 - 63*n + 120)*x^3 + (54*n - 198)*x^4 + 72*x^5)*D[P[n - 2, x], {x, 2}] + (9*n - 35 + (6*n^2 - 47*n + 128)*x + (30*n - 122)*x^2 + 24*x^3)*P[n - 1, x] + ((13*n - 52)*x - (3*n - 64)*x^2 - 12*x^3)*D[P[n - 1, x], x] + ((3*n - 12)*x^2 + 12*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] = (-(8*n - 24 + (6*n^2 - 47*n + 119)*x + (9*n^3 - 96*n^2 + 327*n - 404)*x^2 + (54*n^2 - 360*n + 474)*x^3 + (72*n - 120)*x^4)*v[i - 2] - ((13*n - 39)*x + (30*n^2 - 213*n + 421)*x^2 - (9*n^2 - 225*n + 726)*x^3 - (54*n - 366)*x^4 - 72*x^5)*deriv(v[i - 2], x) - ((3*n - 9)*x^2 + (9*n^2 - 63*n + 120)*x^3 + (54*n - 198)*x^4 + 72*x^5)*deriv(deriv(v[i - 2], x), x) + (9*n - 35 + (6*n^2 - 47*n + 128)*x + (30*n - 122)*x^2 + 24*x^3)*v[i - 1] + ((13*n - 52)*x - (3*n - 64)*x^2 - 12*x^3)*deriv(v[i - 1], x) + ((3*n - 12)*x^2 + 12*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,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.?
  - 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.