A269945 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,x - 4\,n x + n^{2} x\right)\,P_{n-1}(x) + \left(5\,x^{2} - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1	1														1
2	1														1
3	1	1													2
4	1	5	1												7
5	1	14	21	1											37
6	1	30	147	85	1										264
7	1	55	627	1408	341	1									2433
8	1	91	2002	11440	13013	1365	1								27913
9	1	140	5278	61490	196053	118482	5461	1							386906
10	1	204	12138	251498	1733303	3255330	1071799	21845	1						6346119
11	1	285	25194	846260	10787231	46587905	53157079	9668036	87381	1					121159373
12	1	385	48279	2458676	52253971	434928221	1217854704	860181300	87099705	349525	1				2655174768
13	1	506	86779	6369275	209609235	2995372800	16875270660	31306548900	13850000505	784246870	1398101	1			66028903633
14	1	650	148005	15047175	725520510	16410363840	163648537860	638816292660	796513723005	222384254950	7059619931	5592405	1		1845579100993
15	1	819	241605	32955780	2230238010	75177525150	1213911823620	8657594647800	23793900258765	20135227330075	3565207699131	63542171784	22369621	1	57506847262162

Technical details

Node id: n-be6df4e2aaa08cb9 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-de3a1cd8e58b208d

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1;
   3: 1,   1;
   4: 1,   5,     1;
   5: 1,  14,    21,      1;
   6: 1,  30,   147,     85,       1;
   7: 1,  55,   627,   1408,     341,       1;
   8: 1,  91,  2002,  11440,   13013,    1365,       1;
   9: 1, 140,  5278,  61490,  196053,  118482,    5461,     1;
  10: 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n^2-4*n+4)*x)*P(n-1,x) - (2*n-5)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n^2 - 4*n + 4)*x)*P[n - 1, x] - (2*n - 5)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - 4*n + 4)*x)*v[i - 1] - (2*n - 5)*x^2*deriv(v[i - 1], x) + 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, 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) =…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.