A316649 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 24\,x - 18\,x^{2} + 12\,n + 36\,n x + 27\,n x^{2} - 4\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 42\,x + 36\,x^{2} - 30\,n - 105\,n x - 90\,n x^{2} + 12\,n^{2} + 42\,n^{2} x + 36\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 15\,x + 18\,x^{2}\)

\(P_{2}(x) = 26 + 225\,x + 711\,x^{2} + 972\,x^{3} + 486\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	15	18													36
2	26	225	711	972	486											2420
3	252	3096	15579	41121	60102	46170	14580									180900
4	2568	40860	280962	1090989	2618001	3977910	3739770	1990170	459270							14200500
5	26928	526032	4580712	23423472	77917707	176249115	274655124	291269034	201252114	81841914	14880348					1146622500
6	287648	6661200	70164288	444621960	1888318170	5663924433	12306201615	19520536518	22441769280	18240826590	9952472754	3273676560	491051484			94300512500
7	3112896	83362560	1029909456	7781957712	40183597476	150033369096	417787476075	881610018381	1416940407612	1726430020956	1570042169298	1033604915310	465727257468	128585338596	16415149608	7856258062500

All 15 rows

Technical details

Node id: n-207567e2aa72e118 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-25b337cc6c0a5b46

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,          15,           18;
  2:        26,         225,          711,           972,            486;
  3:       252,        3096,        15579,         41121,          60102,          46170,           14580;
  4:      2568,       40860,       280962,       1090989,        2618001,        3977910,         3739770,         1990170,           459270;
  5:     26928,      526032,      4580712,      23423472,       77917707,      176249115,       274655124,       291269034,        201252114,         81841914,         14880348;
  6:    287648,     6661200,     70164288,     444621960,     1888318170,     5663924433,     12306201615,     19520536518,      22441769280,      18240826590,       9952472754,        3273676560,         491051484;
  7:   3112896,    83362560,   1029909456,    7781957712,    40183597476,   150033369096,    417787476075,    881610018381,    1416940407612,    1726430020956,    1570042169298,     1033604915310,      465727257468,     128585338596,      16415149608;
  8:  34013312,  1034064576,  14656159584,  128567836656,   781409312712,  3489593151492,  11846456011026,  31189980384549,   64373878213281,  104515498710252,  133059803969736,   131455873579338,    98812080393642,   54637825470228,   20962146049416,    4986101693430,    554011299270;
  9: 374416128, 12738988800, 203701485312, 2034299556864, 14220017319456, 73891598887008, 295840988467920, 933250753183392, 2351768529036339, 4770160158570471, 7807171270773294, 10287165484508220, 10833078522490686, 8994580456924062, 5759326372281768, 2744489900835540, 916642473047730, 191503239114330, 18836384175180;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(12*n^2-36*n+24)*x+(9*n^2-27*n+18)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(42*n^2-105*n+42)*x+(36*n^2-90*n+36)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2, P(2,x) = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 15*x + 18*x^2;
P[2, x_] := 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*x^2)*P[n - 1, x])/(n^2 - 2*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] = 3 + 15*x + 18*x^2; v[3] = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*x^2)*v[i - 1])/(n^2 - 2*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 in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003