A119308 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-8\,x^{2} + 4\,n x^{2}\right)\,P_{n-3}(x) + \left(2 + 2\,x - 4\,n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(7 + 6\,x + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 9 + 14\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	9	14	4						27
3	28	70	48	8					154
4	90	312	352	144	16				914
5	297	1320	2100	1440	400	32			5589
6	1001	5434	11220	11000	5200	1056	64		34975
7	3432	22022	56056	72072	49280	17248	2688	128	222926

All 15 rows

Technical details

Node id: n-eaddbafbdfb5ca13 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-a10e516e57874c56

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:     3,      2;
  2:     9,     14,       4;
  3:    28,     70,      48,       8;
  4:    90,    312,     352,     144,      16;
  5:   297,   1320,    2100,    1440,     400,      32;
  6:  1001,   5434,   11220,   11000,    5200,    1056,     64;
  7:  3432,  22022,   56056,   72072,   49280,   17248,   2688,    128;
  8: 11934,  88400,  267904,  428064,  387296,  198016,  53760,   6656,   256;
  9: 41990, 352716, 1241136, 2376192, 2690688, 1834560, 733824, 159744, 16128, 512;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = (4*n-8)*x^2*P(n-3,x) - (4*n-2+(4*n-2)*x+4*n*x^2)*P(n-2,x) + (5*n+7+(4*n+6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 9 + 14*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 9 + 14*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8)*x^2*P[n - 3, x] - (4*n - 2 + (4*n - 2)*x + 4*n*x^2)*P[n - 2, x] + (5*n + 7 + (4*n + 6)*x)*P[n - 1, x])/(n + 3)]];
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] = 3 + 2*x; v[3] = 9 + 14*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8)*x^2*v[i - 3] - (4*n - 2 + (4*n - 2)*x + 4*n*x^2)*v[i - 2] + (5*n + 7 + (4*n + 6)*x)*v[i - 1])/(n + 3)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle for first differences of Catalan numbers.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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.