A146988: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.

A146988 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A146988

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 15\,x + 5\,n + 5\,n x\right)\,P_{n-1}(x) + \left(12 + 24\,x + 12\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 1 + 6\,x + x^{2}\)

\(P_{3}(x) = 1 + 19\,x + 19\,x^{2} + x^{3}\)

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	6	1													8
3	1	19	19	1												40
4	1	68	134	68	1											272
5	1	261	778	778	261	1										2080
6	1	1030	4111	6164	4111	1030	1									16448
7	1	4103	20501	40995	40995	20501	4103	1								131200
8	1	16392	98332	245816	327750	245816	98332	16392	1							1048832
9	1	65545	458788	1376340	2293886	2293886	1376340	458788	65545	1						8389120
10	1	262154	2097197	7340152	14680274	18350332	14680274	7340152	2097197	262154	1					67109888
11	1	1048587	9437239	37748901	88080714	132121038	132121038	88080714	37748901	9437239	1048587	1				536872960
12	1	4194316	41943106	188743900	503316975	880804632	1056965532	880804632	503316975	188743900	41943106	4194316	1			4294971392
13	1	16777229	184549454	922747166	2768241355	5536482567	7751075508	7751075508	5536482567	2768241355	922747166	184549454	16777229	1		34359746560
14	1	67108878	805306459	4429185388	14763951081	33218889682	53150223291	62008593768	53150223291	33218889682	14763951081	4429185388	805306459	67108878	1	274877923328

b-files: A146988, A146988 (reversed triangle) ?

Technical details

Node id: n-3f596f3a08c8e69f · revision 0 · JSON
Definition source: oeis_catalog:A146988
Fingerprint: f6-771cff9e4c79260d

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 ?

A146988 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A146988 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,     6,      1;
  3: 1,    19,     19,       1;
  4: 1,    68,    134,      68,       1;
  5: 1,   261,    778,     778,     261,       1;
  6: 1,  1030,   4111,    6164,    4111,    1030,       1;
  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;
  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;
  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 6*x + x^2;
P[3, x_] := 1 + 19*x + 19*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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

A146988 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,     1;
  2: 1,     6,      1;
  3: 1,    19,     19,       1;
  4: 1,    68,    134,      68,       1;
  5: 1,   261,    778,     778,     261,       1;
  6: 1,  1030,   4111,    6164,    4111,    1030,       1;
  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;
  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;
  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 6*x + x^2;
P[3, x_] := 1 + 19*x + 19*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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

A146988 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,     1;
  2: 1,     6,      1;
  3: 1,    19,     19,       1;
  4: 1,    68,    134,      68,       1;
  5: 1,   261,    778,     778,     261,       1;
  6: 1,  1030,   4111,    6164,    4111,    1030,       1;
  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;
  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;
  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 6*x + x^2;
P[3, x_] := 1 + 19*x + 19*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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

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