A166961 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 3\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	9	1						11
3	1	42	59	1					103
4	1	154	925	361	1				1442
5	1	507	8937	16402	2175	1			28023
6	1	1587	67500	365050	265605	13061	1		712805
7	1	4852	442242	5592850	12611845	4127746	78379	1	22857916

All 15 rows

Technical details

Node id: n-9e91ddd3170293d3 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-f600bd83d36e9825

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: 1,     1;
  2: 1,     9,        1;
  3: 1,    42,       59,         1;
  4: 1,   154,      925,       361,          1;
  5: 1,   507,     8937,     16402,       2175,           1;
  6: 1,  1587,    67500,    365050,     265605,       13061,           1;
  7: 1,  4852,   442242,   5592850,   12611845,     4127746,       78379,         1;
  8: 1, 14676,  2652742,  68337922,  365184855,   398536866,    62935117,    470289,       1;
  9: 1, 44181, 15024830, 719764976, 7796966547, 20817224001, 11977188769, 951081090, 2821751, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n^2-n)*x)*P(n-1,x) + (2*x-(4*n-3)*x^2)*d/dx P(n-1,x) + 2*x^3*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n^2 - n)*x)*P[n - 1, x] + (2*x - (4*n - 3)*x^2)*D[P[n - 1, x], x] + 2*x^3*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n^2 - n)*x)*v[i - 1] + (2*x - (4*n - 3)*x^2)*deriv(v[i - 1], x) + 2*x^3*deriv(deriv(v[i - 1], x), 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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

Related ?

Same fingerprint ?

A166961: 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…

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.