A153518: Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

A153518 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153518

\(\left(-3 + n\right)\,P_{n}(x) = \left(-3 - 3\,x + n + n x\right)\,P_{n-1}(x) + \left(-15\,x + 5\,n x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 5 + 5\,x\)

\(P_{2}(x) = 2 + 46\,x + 2\,x^{2}\)

\(P_{3}(x) = 2 + 123\,x + 123\,x^{2} + 2\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	5	5														10
2	2	46	2													50
3	2	123	123	2												250
4	2	135	476	135	2											750
5	2	147	1226	1226	147	2										2750
6	2	159	2048	4832	2048	159	2									9250
7	2	171	2942	13010	13010	2942	171	2								32250
8	2	183	3908	26192	50180	26192	3908	183	2							110750
9	2	195	4946	44810	141422	141422	44810	4946	195	2						382750
10	2	207	6056	69296	317192	533744	317192	69296	6056	207	2					1319250
11	2	219	7238	100082	610538	1558046	1558046	610538	100082	7238	219	2				4552250
12	2	231	8492	137600	1057100	3754544	5784812	3754544	1057100	137600	8492	231	2			15700750
13	2	243	9818	182282	1695110	7864334	17329586	17329586	7864334	1695110	182282	9818	243	2		54162750
14	2	255	11216	234560	2565392	14844944	43966640	63583232	43966640	14844944	2565392	234560	11216	255	2	186829250

b-files: A153518, A153518 (reversed triangle) ?

Technical details

Node id: n-a44eea8c1f4cf4e5 · revision 0 · JSON
Definition source: oeis_catalog:A153518
Fingerprint: f6-e3edd843f2595689

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): no

Matches ?

A153518 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A153518 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Submission text ?

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

%e Example

Triangle begins:
  0: 2;
  1: 5,   5;
  2: 2,  46,    2;
  3: 2, 123,  123,     2;
  4: 2, 135,  476,   135,      2;
  5: 2, 147, 1226,  1226,    147,      2;
  6: 2, 159, 2048,  4832,   2048,    159,     2;
  7: 2, 171, 2942, 13010,  13010,   2942,   171,    2;
  8: 2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
  9: 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (5*n-15)*x*P(n-2,x) for n >= 4, with P(0,x) = 2, P(1,x) = 5 + 5*x, P(2,x) = 2 + 46*x + 2*x^2, P(3,x) = 2 + 123*x + 123*x^2 + 2*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 5 + 5*x;
P[2, x_] := 2 + 46*x + 2*x^2;
P[3, x_] := 2 + 123*x + 123*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (5*n - 15)*x*P[n - 2, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 5*x; v[3] = 2 + 46*x + 2*x^2; v[4] = 2 + 123*x + 123*x^2 + 2*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (5*n - 15)*x*v[i - 2])/(n - 3)); 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

A153518 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 2;
   2: 5,   5;
   3: 2,  46,    2;
   4: 2, 123,  123,     2;
   5: 2, 135,  476,   135,      2;
   6: 2, 147, 1226,  1226,    147,      2;
   7: 2, 159, 2048,  4832,   2048,    159,     2;
   8: 2, 171, 2942, 13010,  13010,   2942,   171,    2;
   9: 2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
  10: 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

%F Formula

The row polynomials satisfy (n-4)*P(n,x) = (n-4+(n-4)*x)*P(n-1,x) + (5*n-20)*x*P(n-2,x) for n >= 5, with P(1,x) = 2, P(2,x) = 5 + 5*x, P(3,x) = 2 + 46*x + 2*x^2, P(4,x) = 2 + 123*x + 123*x^2 + 2*x^3.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 5 + 5*x;
P[3, x_] := 2 + 46*x + 2*x^2;
P[4, x_] := 2 + 123*x + 123*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n - 4 + (n - 4)*x)*P[n - 1, x] + (5*n - 20)*x*P[n - 2, x])/(n - 4)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 5 + 5*x; v[3] = 2 + 46*x + 2*x^2; v[4] = 2 + 123*x + 123*x^2 + 2*x^3; for(n = 5, N, my(i = n); v[i] = ((n - 4 + (n - 4)*x)*v[i - 1] + (5*n - 20)*x*v[i - 2])/(n - 4)); 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

A153518 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 2;
   2: 5,   5;
   3: 2,  46,    2;
   4: 2, 123,  123,     2;
   5: 2, 135,  476,   135,      2;
   6: 2, 147, 1226,  1226,    147,      2;
   7: 2, 159, 2048,  4832,   2048,    159,     2;
   8: 2, 171, 2942, 13010,  13010,   2942,   171,    2;
   9: 2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
  10: 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-4)*P(n,x) = (n-4+(n-4)*x)*P(n-1,x) + (5*n-20)*x*P(n-2,x) for n >= 5, with P(1,x) = 2, P(2,x) = 5 + 5*x, P(3,x) = 2 + 46*x + 2*x^2, P(4,x) = 2 + 123*x + 123*x^2 + 2*x^3.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 5 + 5*x;
P[3, x_] := 2 + 46*x + 2*x^2;
P[4, x_] := 2 + 123*x + 123*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n - 4 + (n - 4)*x)*P[n - 1, x] + (5*n - 20)*x*P[n - 2, x])/(n - 4)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 5 + 5*x; v[3] = 2 + 46*x + 2*x^2; v[4] = 2 + 123*x + 123*x^2 + 2*x^3; for(n = 5, N, my(i = n); v[i] = ((n - 4 + (n - 4)*x)*v[i - 1] + (5*n - 20)*x*v[i - 2])/(n - 4)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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 with a lag row, 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 with a lag row, 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 with a lag row, Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 3
by Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 3
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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

Used in 5

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