A153656: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2j +3)prime(j)*T(n-2, k-1) with j=9, read by rows.

A153656 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153656

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-966\,x + 483\,n x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 23 + 23\,x\)

\(P_{2}(x) = 2 + 1054\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	23	23														46
2	2	1054	2													1058
3	2	12165	12165	2												24334
4	2	13133	533412	13133	2											559682
5	2	14101	6422240	6422240	14101	2										12872686
6	2	15069	12779580	270482476	12779580	15069	2									296071778
7	2	16037	19605432	3385203976	3385203976	19605432	16037	2								6809650894
8	2	17005	26899796	9577346548	137413443860	9577346548	26899796	17005	2							156621970562
9	2	17973	34662672	19073670000	1782044310816	1782044310816	19073670000	34662672	17973	2						3602305322926
10	2	18941	42894060	32100934140	6426976363500	69934782006012	6426976363500	32100934140	42894060	18941	2					82853022427298
11	2	19909	51593960	48885898776	15671659907640	937089160493640	937089160493640	15671659907640	48885898776	51593960	19909	2				1905619515827854
12	2	20877	60762372	69655323716	31225296996036	4056990403971780	35652678029891076	4056990403971780	31225296996036	69655323716	60762372	20877	2			43829248864040642
13	2	21845	70399296	94635968768	54906841428560	11657627436357936	492323732952290976	492323732952290976	11657627436357936	54906841428560	94635968768	70399296	21845	2		1008072723872934766
14	2	22813	80504732	124054593740	88644998752156	26794352726871884	2463507725507018652	18204890954341971660	2463507725507018652	26794352726871884	88644998752156	124054593740	80504732	22813	2	23185672649077499618

b-files: A153656, A153656 (reversed triangle) ?

Technical details

Node id: n-6e554ebca19f83a9 · revision 0 · JSON
Definition source: oeis_catalog:A153656
Fingerprint: f6-86de729c02373aa4

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 ?

A153656 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A153656 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: 23,    23;
  2:  2,  1054,        2;
  3:  2, 12165,    12165,           2;
  4:  2, 13133,   533412,       13133,             2;
  5:  2, 14101,  6422240,     6422240,         14101,             2;
  6:  2, 15069, 12779580,   270482476,      12779580,         15069,           2;
  7:  2, 16037, 19605432,  3385203976,    3385203976,      19605432,       16037,        2;
  8:  2, 17005, 26899796,  9577346548,  137413443860,    9577346548,    26899796,    17005,     2;
  9:  2, 17973, 34662672, 19073670000, 1782044310816, 1782044310816, 19073670000, 34662672, 17973, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (483*n-966)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 23 + 23*x, P(2,x) = 2 + 1054*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 23 + 23*x;
P[2, x_] := 2 + 1054*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (483*n - 966)*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] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (483*n - 966)*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

A153656 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 23,    23;
   3:  2,  1054,        2;
   4:  2, 12165,    12165,           2;
   5:  2, 13133,   533412,       13133,             2;
   6:  2, 14101,  6422240,     6422240,         14101,             2;
   7:  2, 15069, 12779580,   270482476,      12779580,         15069,           2;
   8:  2, 16037, 19605432,  3385203976,    3385203976,      19605432,       16037,        2;
   9:  2, 17005, 26899796,  9577346548,  137413443860,    9577346548,    26899796,    17005,     2;
  10:  2, 17973, 34662672, 19073670000, 1782044310816, 1782044310816, 19073670000, 34662672, 17973, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (483*n-1449)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 23 + 23*x, P(3,x) = 2 + 1054*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 23 + 23*x;
P[3, x_] := 2 + 1054*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (483*n - 1449)*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] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (483*n - 1449)*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

A153656 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  2;
   2: 23,    23;
   3:  2,  1054,        2;
   4:  2, 12165,    12165,           2;
   5:  2, 13133,   533412,       13133,             2;
   6:  2, 14101,  6422240,     6422240,         14101,             2;
   7:  2, 15069, 12779580,   270482476,      12779580,         15069,           2;
   8:  2, 16037, 19605432,  3385203976,    3385203976,      19605432,       16037,        2;
   9:  2, 17005, 26899796,  9577346548,  137413443860,    9577346548,    26899796,    17005,     2;
  10:  2, 17973, 34662672, 19073670000, 1782044310816, 1782044310816, 19073670000, 34662672, 17973, 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) + (483*n-1449)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 23 + 23*x, P(3,x) = 2 + 1054*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 23 + 23*x;
P[3, x_] := 2 + 1054*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (483*n - 1449)*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] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (483*n - 1449)*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

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