A088617 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(3 - n\right)\,P_{n-2}(x) + \left(-9 - 24\,x + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	6	28	32						66
4	22	164	400	320					906
5	90	924	3504	5824	3584				13926
6	394	5156	26656	68096	86016	43008			229326
7	1806	28732	188480	652800	1259520	1284096	540672		3956106
8	8558	160324	1275696	5590080	14572800	22606848	19329024	7028736	70572066

All 15 rows

Technical details

Node id: n-39310b120a1d8578 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2eec2e552ea6c0fc

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      2,       4;
   3:      6,      28,       32;
   4:     22,     164,      400,       320;
   5:     90,     924,     3504,      5824,       3584;
   6:    394,    5156,    26656,     68096,      86016,      43008;
   7:   1806,   28732,   188480,    652800,    1259520,    1284096,     540672;
   8:   8558,  160324,  1275696,   5590080,   14572800,   22606848,   19329024,    7028736;
   9:  41586,  896572,  8390480,  44524480,  146558720,  306482176,  397709312,  292864000,   93716480;
  10: 206098, 5025860, 54091840, 337272320, 1342781440, 3540432896, 6182944768, 6897532928, 4460904448, 1274544128;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-3)*P(n-2,x) + (6*n-9+(16*n-24)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 3)*P[n - 2, x] + (6*n - 9 + (16*n - 24)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n - 3)*v[i - 2] + (6*n - 9 + (16*n - 24)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

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