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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	10	79	273	539	665	525	259	73	9							2432
9	11	98	388	896	1330	1316	868	368	91	10						5376
10	12	119	531	1404	2436	2898	2394	1356	504	111	11					11776
11	13	142	705	2100	4170	5796	5754	4080	2025	670	133	12				25600
12	14	167	913	3025	6765	10758	12474	10626	6600	2915	869	157	13			55296
13	15	194	1158	4224	10505	18810	24948	24816	18513	10230	4070	1104	183	14		118784
14	16	223	1443	5746	15730	31317	46761	53196	46332	30745	15301	5538	1378	211	15	253952

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