A059110 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 2\,x^{2} + 6\,x^{3} + 2\,n x - 3\,n x^{2} - 5\,n x^{3} + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} + 4\,x^{3} + 6\,x^{4} - 2\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{4} + x^{5}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	1	4	3						8
4	1	9	21	13					44
5	1	16	78	136	73				304
6	1	25	210	730	1045	501			2512
7	1	36	465	2720	7515	9276	4051		24064
8	1	49	903	8015	36575	85071	93289	37633	261536

All 15 rows

Technical details

Node id: n-314823098ef7e229 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-65341f67bf7e02f3

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: 1,  1;
   3: 1,  4,    3;
   4: 1,  9,   21,    13;
   5: 1, 16,   78,   136,     73;
   6: 1, 25,  210,   730,   1045,     501;
   7: 1, 36,  465,  2720,   7515,    9276,    4051;
   8: 1, 49,  903,  8015,  36575,   85071,   93289,    37633;
   9: 1, 64, 1596, 20048, 137270,  519456, 1053724,  1047376,   394353;
  10: 1, 81, 2628, 44436, 427518, 2404206, 7836276, 14196708, 12975561, 4596553;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-3)*x+(n^2-3*n+2)*x^2+(n^2-5*n+6)*x^3)*P(n-1,x) - (2*x^2+(2*n-4)*x^3+(2*n-6)*x^4)*d/dx P(n-1,x) + (x^4+x^5)*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 + (2*n - 3)*x + (n^2 - 3*n + 2)*x^2 + (n^2 - 5*n + 6)*x^3)*P[n - 1, x] - (2*x^2 + (2*n - 4)*x^3 + (2*n - 6)*x^4)*D[P[n - 1, x], x] + (x^4 + x^5)*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 + (2*n - 3)*x + (n^2 - 3*n + 2)*x^2 + (n^2 - 5*n + 6)*x^3)*v[i - 1] - (2*x^2 + (2*n - 4)*x^3 + (2*n - 6)*x^4)*deriv(v[i - 1], x) + (x^4 + x^5)*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 T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n an…?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A059110: Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.

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.