A153651: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)prime(j)T(n-2, k-1) with j=6, read by rows.

A153651 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153651

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-286\,x + 143\,n x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 13 + 13\,x\)

\(P_{2}(x) = 2 + 334\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	13	13														26
2	2	334	2													338
3	2	2195	2195	2												4394
4	2	2483	52152	2483	2											57122
5	2	2771	368520	368520	2771	2										742586
6	2	3059	726360	8194776	726360	3059	2									9653618
7	2	3347	1125672	61619496	61619496	1125672	3347	2								125497034
8	2	3635	1566456	166614648	1295091960	166614648	1566456	3635	2							1631461442
9	2	3923	2048712	329152200	10273294536	10273294536	329152200	2048712	3923	2						21208998746
10	2	4211	2572440	555204120	34428341400	205744739352	34428341400	555204120	2572440	4211	2					275716983698
11	2	4499	3137640	850742376	82052310120	1709254199400	1709254199400	82052310120	850742376	3137640	4499	2				3584320788074
12	2	4787	3744312	1221738936	162297241656	6714559329720	32840006126136	6714559329720	162297241656	1221738936	3744312	4787	2			46596170244962
13	2	5075	4392456	1674165768	285175140360	18610336918536	283977915970056	283977915970056	18610336918536	285175140360	1674165768	4392456	5075	2		605750213184506
14	2	5363	5082072	2213994840	461557973976	42104017615704	1262770237038552	5264076707977560	1262770237038552	42104017615704	461557973976	2213994840	5082072	5363	2	7874752771398578

b-files: A153651, A153651 (reversed triangle) ?

Technical details

Node id: n-2a67b0774691ba3b · revision 0 · JSON
Definition source: oeis_catalog:A153651
Fingerprint: f6-aed9b872eb24ac87

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 ?

A153651 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A153651 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

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

%e Example

Triangle begins:
  0:  2;
  1: 13,   13;
  2:  2,  334,       2;
  3:  2, 2195,    2195,         2;
  4:  2, 2483,   52152,      2483,           2;
  5:  2, 2771,  368520,    368520,        2771,           2;
  6:  2, 3059,  726360,   8194776,      726360,        3059,         2;
  7:  2, 3347, 1125672,  61619496,    61619496,     1125672,      3347,       2;
  8:  2, 3635, 1566456, 166614648,  1295091960,   166614648,   1566456,    3635,    2;
  9:  2, 3923, 2048712, 329152200, 10273294536, 10273294536, 329152200, 2048712, 3923, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (143*n-286)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 13 + 13*x, P(2,x) = 2 + 334*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 13 + 13*x;
P[2, x_] := 2 + 334*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (143*n - 286)*x*P[n - 2, x])/(n - 2)]];
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] = 13 + 13*x; v[3] = 2 + 334*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (143*n - 286)*x*v[i - 2])/(n - 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

A153651 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 13,   13;
   3:  2,  334,       2;
   4:  2, 2195,    2195,         2;
   5:  2, 2483,   52152,      2483,           2;
   6:  2, 2771,  368520,    368520,        2771,           2;
   7:  2, 3059,  726360,   8194776,      726360,        3059,         2;
   8:  2, 3347, 1125672,  61619496,    61619496,     1125672,      3347,       2;
   9:  2, 3635, 1566456, 166614648,  1295091960,   166614648,   1566456,    3635,    2;
  10:  2, 3923, 2048712, 329152200, 10273294536, 10273294536, 329152200, 2048712, 3923, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (143*n-429)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 13 + 13*x, P(3,x) = 2 + 334*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 13 + 13*x;
P[3, x_] := 2 + 334*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (143*n - 429)*x*P[n - 2, x])/(n - 3)]];
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] = 13 + 13*x; v[3] = 2 + 334*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (143*n - 429)*x*v[i - 2])/(n - 3)); 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

A153651 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 13,   13;
   3:  2,  334,       2;
   4:  2, 2195,    2195,         2;
   5:  2, 2483,   52152,      2483,           2;
   6:  2, 2771,  368520,    368520,        2771,           2;
   7:  2, 3059,  726360,   8194776,      726360,        3059,         2;
   8:  2, 3347, 1125672,  61619496,    61619496,     1125672,      3347,       2;
   9:  2, 3635, 1566456, 166614648,  1295091960,   166614648,   1566456,    3635,    2;
  10:  2, 3923, 2048712, 329152200, 10273294536, 10273294536, 329152200, 2048712, 3923, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (143*n-429)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 13 + 13*x, P(3,x) = 2 + 334*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 13 + 13*x;
P[3, x_] := 2 + 334*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (143*n - 429)*x*P[n - 2, x])/(n - 3)]];
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] = 13 + 13*x; v[3] = 2 + 334*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (143*n - 429)*x*v[i - 2])/(n - 3)); 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 for three-term recurrences, Liu–Wang theorem with a lag row
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, Liu–Wang theorem with a lag row
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
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 ≥ 2
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
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check

Used in 4

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