A360282 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 12\,x - 18\,x^{2} + 5\,n + 30\,n x + 45\,n x^{2} - 2\,n^{2} - 12\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 12\,x - 24\,n - 24\,n x + 12\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	5	21	18													44
3	25	129	189	81												424
4	129	783	1566	1242	324											4044
5	681	4743	11880	13500	6885	1215										38904
6	3653	28689	85860	126090	95175	34263	4374									378104
7	19825	173253	601209	1078245	1074465	586845	158193	15309								3707344
8	108545	1044507	4115412	8696268	10744650	7875630	3286332	691092	52488							36614924
9	598417	6286539	27689688	67250736	99018612	91057932	51683184	17111088	2893401	177147						363766744
10	3317445	37775565	183781710	503785980	860127660	951056316	683883648	311516280	84144825	11711385	590490					3631691304
11	18474633	226647369	1206320445	3680716545	7143057900	9223100964	8033979492	4675237380	1754837865	395136225	46117269	1948617				36405574704
12	103274625	1357926183	7845196194	26352902070	57255365970	84510557010	86472212904	61350546312	29635216875	9354444165	1786232250	177501294	6377292			366207753144
13	579168825	8125177167	50621652432	185545250748	445944515445	740368970055	870163052616	728916707376	431716745889	176521966335	47624199480	7820685756	670147101	20726199		3694638965424
14	3256957317	48558369609	324441503568	1288075070646	3391748527167	6253805989431	8300446996842	8020181647764	5633945613153	2838900485421	997980299316	233197373682	33320819259	2488738203	66961566	37370415352944

Technical details

Node id: n-8d61092fcd4f7d86 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-7a00faae32479261

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:      1,       3;
  2:      5,      21,       18;
  3:     25,     129,      189,       81;
  4:    129,     783,     1566,     1242,      324;
  5:    681,    4743,    11880,    13500,     6885,     1215;
  6:   3653,   28689,    85860,   126090,    95175,    34263,     4374;
  7:  19825,  173253,   601209,  1078245,  1074465,   586845,   158193,    15309;
  8: 108545, 1044507,  4115412,  8696268, 10744650,  7875630,  3286332,   691092,   52488;
  9: 598417, 6286539, 27689688, 67250736, 99018612, 91057932, 51683184, 17111088, 2893401, 177147;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(12*n^2-30*n+12)*x+(18*n^2-45*n+18)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(12*n^2-24*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (12*n^2 - 30*n + 12)*x + (18*n^2 - 45*n + 18)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (12*n^2 - 24*n + 12)*x)*P[n - 1, x])/(2*n^2 - 3*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (12*n^2 - 30*n + 12)*x + (18*n^2 - 45*n + 18)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (12*n^2 - 24*n + 12)*x)*v[i - 1])/(2*n^2 - 3*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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
  - 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
  - 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

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.