A123097: Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.

A123097 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A123097

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2 + 4\,x + 2\,x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 4 + 7\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	4	7	3						14
3	5	14	13	4					36
4	6	23	33	21	5				88
5	7	34	66	64	31	6			208
6	8	47	115	150	110	43	7		480
7	9	62	183	300	295	174	57	8	1088

All 15 rows b-files: A052951 (row sums), A123097 ?

Technical details

Node id: n-e48eea24800097e1 · revision 0 · JSON
Definition source: oeis_catalog:A123097
Fingerprint: f6-38f9189b2b86204b

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: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A123097 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A052951

Submission text ?

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

%e Example

Triangle begins:
  0:  1;
  1:  3,  2;
  2:  4,  7,   3;
  3:  5, 14,  13,   4;
  4:  6, 23,  33,  21,    5;
  5:  7, 34,  66,  64,   31,    6;
  6:  8, 47, 115, 150,  110,   43,   7;
  7:  9, 62, 183, 300,  295,  174,  57,   8;
  8: 10, 79, 273, 539,  665,  525, 259,  73,  9;
  9: 11, 98, 388, 896, 1330, 1316, 868, 368, 91, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 4 + 7*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 4 + 7*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*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] = 1; v[2] = 3 + 2*x; v[3] = 4 + 7*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*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

A123097 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:  1;
   2:  3,  2;
   3:  4,  7,   3;
   4:  5, 14,  13,   4;
   5:  6, 23,  33,  21,    5;
   6:  7, 34,  66,  64,   31,    6;
   7:  8, 47, 115, 150,  110,   43,   7;
   8:  9, 62, 183, 300,  295,  174,  57,   8;
   9: 10, 79, 273, 539,  665,  525, 259,  73,  9;
  10: 11, 98, 388, 896, 1330, 1316, 868, 368, 91, 10;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(2*n-6)*x)*P(n-1,x) - (n-3+(2*n-6)*x+(n-3)*x^2)*P(n-2,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 4 + 7*x + 3*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 4 + 7*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (2*n - 6)*x)*P[n - 1, x] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*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] = 1; v[2] = 3 + 2*x; v[3] = 4 + 7*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = ((2*n - 6 + (2*n - 6)*x)*v[i - 1] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*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

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

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

Related ?

Same fingerprint ?

A156992 → Pₙ + Pₙ₊₁

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