A209414: Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.

A209414 ↗ nonnegative coefficients

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

Recurrence from A209414

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	7	9	1					18
4	1	10	26	16	1				54
5	1	13	52	70	25	1			162
6	1	16	87	190	155	36	1		486
7	1	19	131	403	553	301	49	1	1458

All 15 rows b-files: A008776 (row sums), A025192 (row sums), A114188 (reversed triangle), A209414 ?

Technical details

Node id: n-522def1e8ebf49cd · revision 0 · JSON
Definition source: oeis_catalog:A209414
Fingerprint: f6-a9c88078fbd18a69

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 ?

A114188 reversed triangle 66 terms, 11 rows · tabl · nonn b-file?
A209414 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A027334, A099856, A025192, A008776, A110593

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   1;
  3: 1,  7,   9,    1;
  4: 1, 10,  26,   16,    1;
  5: 1, 13,  52,   70,   25,    1;
  6: 1, 16,  87,  190,  155,   36,    1;
  7: 1, 19, 131,  403,  553,  301,   49,   1;
  8: 1, 22, 184,  736, 1462, 1372,  532,  64,  1;
  9: 1, 25, 246, 1216, 3206, 4446, 3024, 876, 81, 1;

%F Formula

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

%t Mathematica

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

A114188 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   1;
  3: 1,  9,   7,    1;
  4: 1, 16,  26,   10,    1;
  5: 1, 25,  70,   52,   13,    1;
  6: 1, 36, 155,  190,   87,   16,    1;
  7: 1, 49, 301,  553,  403,  131,   19,   1;
  8: 1, 64, 532, 1372, 1462,  736,  184,  22,  1;
  9: 1, 81, 876, 3024, 4446, 3206, 1216, 246, 25, 1;

%F Formula

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

%t Mathematica

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

A209414 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  4,   1;
   4: 1,  7,   9,    1;
   5: 1, 10,  26,   16,    1;
   6: 1, 13,  52,   70,   25,    1;
   7: 1, 16,  87,  190,  155,   36,    1;
   8: 1, 19, 131,  403,  553,  301,   49,   1;
   9: 1, 22, 184,  736, 1462, 1372,  532,  64,  1;
  10: 1, 25, 246, 1216, 3206, 4446, 3024, 876, 81, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

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

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