A209599: Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A209599 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)basis map preserves real roots

Recurrence from A209599

\(P_{n}(x) = \left(1\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	3	1							4
3	5	3							8
4	8	7	1						16
5	13	15	4						32
6	21	30	12	1					64
7	34	58	31	5					128
8	55	109	73	18	1				256
9	89	201	162	54	6				512
10	144	365	344	145	25	1			1024
11	233	655	707	361	85	7			2048
12	377	1164	1416	850	255	33	1		4096
13	610	2052	2778	1918	701	125	8		8192
14	987	3593	5358	4184	1806	413	42	1	16384

b-files: A000079 (row sums) ?

Technical details

Node id: n-72cbbb2f152c8f9c · revision 0 · JSON
Definition source: oeis_catalog:A209599
Fingerprint: f6-4899378506777ad9

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

Row sums ? A166444, A118655, A220051, A034008, A011782, A050732, A000079, A131577

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  2;
  2:  3,   1;
  3:  5,   3;
  4:  8,   7,   1;
  5: 13,  15,   4;
  6: 21,  30,  12,  1;
  7: 34,  58,  31,  5;
  8: 55, 109,  73, 18, 1;
  9: 89, 201, 162, 54, 6;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + (1 + x)*P[n - 2, x]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + (1 + x)*v[i - 2]); 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 (−∞, -1]
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, -1]: every PF input has a real-rooted image with every root <= -c/b = -1
by Constant-current basis map
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, -1]: every PF input has a PF image (zero allowed)
by Constant-current basis map

Proof chains ? JSON

Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? 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

Related ?

Same fingerprint ?

A236076: A skewed version of triangular array A122075.

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