A109983 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 4\,x + 4\,x^{2} - n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 10\,x - 8\,x^{2} + 6\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 10\,x + 8\,x^{2}\)

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	10	8													21
2	13	88	220	240	96											657
3	63	642	2700	5992	7392	4800	1280									22869
4	321	4368	25824	86592	180016	237440	193920	89600	17920							836001
5	1683	28650	218280	979840	2868880	5722720	7873280	7374080	4497920	1612800	258048					31436181
6	8989	183720	1713324	9638320	36419280	97355520	188743360	267321600	274444800	199127040	96897024	28385280	3784704			1204022961
7	48639	1160194	12800060	86561160	400793008	1343864480	3364482240	6388086400	9241648640	10135802880	8294221824	4909707264	1986969600	492011520	56229888	46714387797

All 15 rows

Technical details

Node id: n-dbf83e3ea48cbbcf · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-1d4b84e819f54819

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,       10,         8;
  2:      13,       88,       220,        240,          96;
  3:      63,      642,      2700,       5992,        7392,         4800,         1280;
  4:     321,     4368,     25824,      86592,      180016,       237440,       193920,         89600,         17920;
  5:    1683,    28650,    218280,     979840,     2868880,      5722720,      7873280,       7374080,       4497920,       1612800,         258048;
  6:    8989,   183720,   1713324,    9638320,    36419280,     97355520,    188743360,     267321600,     274444800,     199127040,       96897024,       28385280,        3784704;
  7:   48639,  1160194,  12800060,   86561160,   400793008,   1343864480,   3364482240,    6388086400,    9241648640,   10135802880,     8294221824,     4909707264,     1986969600,     492011520,      56229888;
  8:  265729,  7245856,  92309920,  729237376,  3997885696,  16126136320,  49501204480,  117939742720,  220391760640,  324045680640,   373583745024,   334109884416,   227205242880,  113556258816,   39332806656,    8434483200,    843448320;
  9: 1462563, 44874954, 648295272, 5858890176, 37114530048, 174989283840, 636330240000, 1824612710400, 4181764942080, 7716832898560, 11491931547648, 13776668319744, 13195012841472, 9958952779776, 5792296992768, 2504760360960, 758260039680, 143386214400, 12745441280;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-1+(4*n-4)*x+(4*n-4)*x^2)*P(n-2,x) + (6*n-3+(20*n-10)*x+(16*n-8)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 10*x + 8*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 10*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*P[n - 2, x] + (6*n - 3 + (20*n - 10)*x + (16*n - 8)*x^2)*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] = 3 + 10*x + 8*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*v[i - 2] + (6*n - 3 + (20*n - 10)*x + (16*n - 8)*x^2)*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) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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
[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.