A124848 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(2 + 3\,x - 2\,n - 3\,n x\right)\,P_{n-2}(x) + \left(2 + 9\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	12							17
2	19	84	90						193
3	63	396	810	540					1809
4	192	1548	4590	5940	2835				15105
5	552	5400	20790	39420	36855	13608			116625
6	1520	17424	82080	203580	280665	204120	61236		850625
7	4048	53088	294840	899640	1630125	1755432	1041012	262440	5940625

All 15 rows

Technical details

Node id: n-35eb54760ab979ef · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0aae4bfdf29522d2

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:     5,     12;
  2:    19,     84,      90;
  3:    63,    396,     810,      540;
  4:   192,   1548,    4590,     5940,     2835;
  5:   552,   5400,   20790,    39420,    36855,    13608;
  6:  1520,  17424,   82080,   203580,   280665,   204120,    61236;
  7:  4048,  53088,  294840,   899640,  1630125,  1755432,  1041012,   262440;
  8: 10496, 154752,  987840,  3568320,  7983360, 11335464,  9981468,  4986360,  1082565;
  9: 26624, 435456, 3136320, 13063680, 34700400, 60991056, 70972524, 52750440, 22733865, 4330260;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n-2+(3*n-3)*x)*P(n-2,x) + (3*n+2+(3*n+9)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (3*n - 3)*x)*P[n - 2, x] + (3*n + 2 + (3*n + 9)*x)*P[n - 1, x])/(n)]];
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] = 5 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 2 + (3*n - 3)*x)*v[i - 2] + (3*n + 2 + (3*n + 9)*x)*v[i - 1])/(n)); 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 read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - 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.