A316649 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2} + 12\,n + 12\,n x + 3\,n x^{2} - 4\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 14\,x + 4\,x^{2} - 30\,n - 35\,n x - 10\,n x^{2} + 12\,n^{2} + 14\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 5\,x + 2\,x^{2}\)

\(P_{2}(x) = 26 + 75\,x + 79\,x^{2} + 36\,x^{3} + 6\,x^{4}\)

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	5	2													10
2	26	75	79	36	6											222
3	252	1032	1731	1523	742	190	20									5490
4	2568	13620	31218	40407	32321	16370	5130	910	70							142614
5	26928	175344	508968	867536	961947	725305	376756	133182	30674	4158	252					3811050
6	287648	2220400	7796032	16467480	23312570	23308331	16880935	8925714	3420480	926730	168546	18480	924			103734270
7	3112896	27787520	114434384	288220656	496093796	617421272	573096675	403113863	215964092	87711732	26588802	5834730	876348	80652	3432	2860340850

All 15 rows

Technical details

Node id: n-451dfe842bdfe4bd · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-fbcf47e112a39ff9

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,          5,           2;
  2:        26,         75,          79,          36,            6;
  3:       252,       1032,        1731,        1523,          742,          190,           20;
  4:      2568,      13620,       31218,       40407,        32321,        16370,         5130,          910,           70;
  5:     26928,     175344,      508968,      867536,       961947,       725305,       376756,       133182,        30674,         4158,          252;
  6:    287648,    2220400,     7796032,    16467480,     23312570,     23308331,     16880935,      8925714,      3420480,       926730,       168546,       18480,         924;
  7:   3112896,   27787520,   114434384,   288220656,    496093796,    617421272,    573096675,    403113863,    215964092,     87711732,     26588802,     5834730,      876348,      80652,       3432;
  8:  34013312,  344688192,  1628462176,  4761771728,   9647028552,  14360465644,  16250282594,  14261536527,   9811595521,   5309937444,   2253379464,   742072254,   185932362,   34270236,    4382664,    347490,    12870;
  9: 374416128, 4246329600, 22633498368, 75344428032, 175555769376, 304080653856, 405817542480, 426726453216, 358446658899, 242349243437, 132215131006, 58071350260, 20384348446, 5641629994, 1204132072, 191268220, 21294130, 1482910, 48620;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(4*n^2-12*n+8)*x+(n^2-3*n+2)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(14*n^2-35*n+14)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 5*x + 2*x^2, P(2,x) = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 5*x + 2*x^2;
P[2, x_] := 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*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 + 5*x + 2*x^2; v[3] = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*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