A146987: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.

A146987 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A146987

\(\left(-3 + n\right)\,P_{n}(x) = \left(-12 - 12\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(9 + 18\,x + 9\,x^{2} - 3\,n - 6\,n x - 3\,n x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 1 + 5\,x + x^{2}\)

\(P_{3}(x) = 1 + 12\,x + 12\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	5	1													7
3	1	12	12	1												26
4	1	31	60	31	1											124
5	1	86	253	253	86	1										680
6	1	249	987	1478	987	249	1									3952
7	1	736	3666	7325	7325	3666	736	1								23456
8	1	2195	13150	32861	43810	32861	13150	2195	1							140224
9	1	6570	45963	137865	229761	229761	137865	45963	6570	1						840320
10	1	19693	157509	551244	1102458	1378062	1102458	551244	157509	19693	1					5039872
11	1	59060	531496	2125929	4960446	7440636	7440636	4960446	2125929	531496	59060	1				30235136
12	1	177159	1771536	7971835	21258135	37201662	44641968	37201662	21258135	7971835	1771536	177159	1			181402624
13	1	531454	5845929	29229541	87688480	175376817	245527458	245527458	175376817	87688480	29229541	5845929	531454	1		1088399360
14	1	1594337	19131967	105225682	350752061	789191887	1262706819	1473157884	1262706819	789191887	350752061	105225682	19131967	1594337	1	6530363392

b-files: A146987, A146987 (reversed triangle) ?

Technical details

Node id: n-fbab854f2622405b · revision 0 · JSON
Definition source: oeis_catalog:A146987
Fingerprint: f6-fe0a8ead5b1a5b27

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 ?

A146987 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A146987 reversed 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: 1;
  1: 1,    1;
  2: 1,    5,     1;
  3: 1,   12,    12,      1;
  4: 1,   31,    60,     31,      1;
  5: 1,   86,   253,    253,     86,      1;
  6: 1,  249,   987,   1478,    987,    249,      1;
  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;
  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;
  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[3, x_] := 1 + 12*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*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] = 1; v[2] = 1 + x; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*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

A146987 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    5,     1;
  3: 1,   12,    12,      1;
  4: 1,   31,    60,     31,      1;
  5: 1,   86,   253,    253,     86,      1;
  6: 1,  249,   987,   1478,    987,    249,      1;
  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;
  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;
  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[3, x_] := 1 + 12*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*P[n - 2, x])/(n - 3)]];
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] = 1 + x; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*v[i - 2])/(n - 3)); 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

A146987 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    5,     1;
  3: 1,   12,    12,      1;
  4: 1,   31,    60,     31,      1;
  5: 1,   86,   253,    253,     86,      1;
  6: 1,  249,   987,   1478,    987,    249,      1;
  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;
  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;
  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[3, x_] := 1 + 12*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*P[n - 2, x])/(n - 3)]];
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; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*v[i - 2])/(n - 3)); 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 ≥ 3
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

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