A249247 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	5	12							17
3	17	48	16						81
4	73	224	112						409
5	382	1236	768	64					2450
6	2365	7932	5616	832					16745
7	16937	58288	45024	8960	256				129465
8	137861	483696	397536	94976	5376				1119445
9	1257686	4479300	3856000	1043840	84480	1024			10722330
10	12714721	45828140	40892320	12123520	1230080	31744			112820525
11	141119617	513619584	471588720	149826560	17790720	688128	4096		1294637425
12	1706150125	6260122032	5883269520	1973611520	263212800	13209600	176128		16099751725
13	22321071242	82459684468	79008570816	27693131200	4038863360	243575808	5046272	16384	215769959550
14	314201147513	1167520305084	1137043749072	413210526400	64701745920	4476122112	123662336	933888	3101278192325

Technical details

Node id: n-11b1315d0a74bf89 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-6479b01c4332fd86

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:       1;
  1:       2,       4;
  2:       5,      12;
  3:      17,      48,      16;
  4:      73,     224,     112;
  5:     382,    1236,     768,      64;
  6:    2365,    7932,    5616,     832;
  7:   16937,   58288,   45024,    8960,   256;
  8:  137861,  483696,  397536,   94976,  5376;
  9: 1257686, 4479300, 3856000, 1043840, 84480, 1024;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 2, x] + n*P[n - 1, x]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 2] + n*v[i - 1]); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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