A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

A134346 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A134346

\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	3							6
2	7	14	7						28
3	15	45	45	15					120
4	31	124	186	124	31				496
5	63	315	630	630	315	63			2016
6	127	762	1905	2540	1905	762	127		8128
7	255	1785	5355	8925	8925	5355	1785	255	32640

All 15 rows b-files: A134346, A134346 (reversed triangle), A171476 (row sums), A171496 (row sums) ?

Technical details

Node id: n-8d96419786c3ee2a · revision 0 · JSON
Definition source: oeis_catalog:A134346
Fingerprint: f6-77a7ed3b1d5f257e

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A134346 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A134346 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A006516, A171496, A171476

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    3,    3;
  2:    7,   14,     7;
  3:   15,   45,    45,    15;
  4:   31,  124,   186,   124,     31;
  5:   63,  315,   630,   630,    315,     63;
  6:  127,  762,  1905,  2540,   1905,    762,   127;
  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;
  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;
  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*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

A134346 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    3,    3;
  2:    7,   14,     7;
  3:   15,   45,    45,    15;
  4:   31,  124,   186,   124,     31;
  5:   63,  315,   630,   630,    315,     63;
  6:  127,  762,  1905,  2540,   1905,    762,   127;
  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;
  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;
  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A134346 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    3,    3;
  2:    7,   14,     7;
  3:   15,   45,    45,    15;
  4:   31,  124,   186,   124,     31;
  5:   63,  315,   630,   630,    315,     63;
  6:  127,  762,  1905,  2540,   1905,    762,   127;
  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;
  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;
  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 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

Proof chains ? JSON

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 ?

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