A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

A049403 (tail) ↗ nonnegative coefficients

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

Recurrence from A049403

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

\(P_{0}(x) = 3\,x + x^{2}\)

\(P_{1}(x) = 3\,x + 6\,x^{2} + x^{3}\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	3	1															4
1	3	6	1														10
2	0	15	10	1													26
3	0	15	45	15	1												76
4	0	0	105	105	21	1											232
5	0	0	105	420	210	28	1										764
6	0	0	0	945	1260	378	36	1									2620
7	0	0	0	945	4725	3150	630	45	1								9496
8	0	0	0	0	10395	17325	6930	990	55	1							35696
9	0	0	0	0	10395	62370	51975	13860	1485	66	1						140152
10	0	0	0	0	0	135135	270270	135135	25740	2145	78	1					568504
11	0	0	0	0	0	135135	945945	945945	315315	45045	3003	91	1				2390480
12	0	0	0	0	0	0	2027025	4729725	2837835	675675	75075	4095	105	1			10349536
13	0	0	0	0	0	0	2027025	16216200	18918900	7567560	1351350	120120	5460	120	1		46206736
14	0	0	0	0	0	0	0	34459425	91891800	64324260	18378360	2552550	185640	7140	136	1	211799312

Technical details

Node id: n-47e3a1a59edf16da · revision 0 · JSON
Definition source: oeis_catalog:A049403
Fingerprint: f6-eccca8d515312e3b

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): no

Matches ?

Row sums ? A000085

Submission text ?

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

%e Example

Triangle begins:
  0: 0, 3,  1;
  1: 0, 3,  6,   1;
  2: 0, 0, 15,  10,   1;
  3: 0, 0, 15,  45,  15,     1;
  4: 0, 0,  0, 105, 105,    21,     1;
  5: 0, 0,  0, 105, 420,   210,    28,     1;
  6: 0, 0,  0,   0, 945,  1260,   378,    36,     1;
  7: 0, 0,  0,   0, 945,  4725,  3150,   630,    45,    1;
  8: 0, 0,  0,   0,   0, 10395, 17325,  6930,   990,   55,  1;
  9: 0, 0,  0,   0,   0, 10395, 62370, 51975, 13860, 1485, 66, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 3*x + x^2;
P[1, x_] := 3*x + 6*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n + 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 3*x + x^2; v[2] = 3*x + 6*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n + 2)*x*v[i - 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

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 ≥ 1
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

Related ?

Same fingerprint ?

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