A146987 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 45\,x + 9\,x^{2} - 12\,n - 15\,n x - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(45\,x + 18\,x^{2} - 15\,n x - 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(9\,x^{2} - 3\,n x^{2}\right)\,P_{n-2}''(x) + \left(-24 - 12\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-12\,x + 4\,n x\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 7 + 8\,x + x^{2}\)

\(P_{3}(x) = 26 + 55\,x + 18\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	7	8	1													16
3	26	55	18	1												100
4	124	443	271	41	1											880
5	680	3937	4011	1178	101	1										9908
6	3952	36143	57241	27883	4862	270	1									130352
7	23456	333025	783183	578166	166405	19388	764	1								1904388
8	140224	3054983	10344601	10950891	4689426	895527	75072	2231	1							30152956
9	840320	27877177	133076871	194888758	116949021	32758131	4466049	283233	6615	1						511146176
10	5039872	253299503	1679622481	3320296643	2687460322	1037640996	205165275	21003198	1044849	19748	1					9210592888
11	30235136	2294464945	20911526223	54831058886	58319180405	29842665826	8033408418	1181907717	94279824	3781501	59126	1				175542568008
12	181402624	20739835223	257805991081	885135328891	1214130193826	801566295849	282291774094	56002182357	6372691017	407501735	13466461	177237	1			3524646840396
13	1088399360	187199552617	3155827470231	14050106522838	24509034961741	20474563970201	9179310034763	2358810259201	359163457785	32562803482	1707091595	47302708	531545	1		74309422358068
14	6530363392	1688051638463	38431269414721	220217309406003	483308320413282	503487201976902	281723282472489	91224299791196	17850972326473	2152007973401	159138406570	6965758257	164221184	1594442	1	1640255515756776

Technical details

Node id: n-919aa614b1337522 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-e10f38a72634eca7

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:      1;
  1:      2,        1;
  2:      7,        8,         1;
  3:     26,       55,        18,         1;
  4:    124,      443,       271,        41,         1;
  5:    680,     3937,      4011,      1178,       101,        1;
  6:   3952,    36143,     57241,     27883,      4862,      270,       1;
  7:  23456,   333025,    783183,    578166,    166405,    19388,     764,      1;
  8: 140224,  3054983,  10344601,  10950891,   4689426,   895527,   75072,   2231,    1;
  9: 840320, 27877177, 133076871, 194888758, 116949021, 32758131, 4466049, 283233, 6615, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(15*n-45)*x+(3*n-9)*x^2)*P(n-2,x) - ((15*n-45)*x+(6*n-18)*x^2)*d/dx P(n-2,x) - (3*n-9)*x^2*d^2/dx^2 P(n-2,x) + (8*n-24+(4*n-12)*x)*P(n-1,x) + (4*n-12)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 8*x + x^2, P(3,x) = 26 + 55*x + 18*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 7 + 8*x + x^2;
P[3, x_] := 26 + 55*x + 18*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (15*n - 45)*x + (3*n - 9)*x^2)*P[n - 2, x] - ((15*n - 45)*x + (6*n - 18)*x^2)*D[P[n - 2, x], x] - (3*n - 9)*x^2*D[P[n - 2, x], {x, 2}] + (8*n - 24 + (4*n - 12)*x)*P[n - 1, x] + (4*n - 12)*x*D[P[n - 1, x], 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] = 2 + x; v[3] = 7 + 8*x + x^2; v[4] = 26 + 55*x + 18*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (15*n - 45)*x + (3*n - 9)*x^2)*v[i - 2] - ((15*n - 45)*x + (6*n - 18)*x^2)*deriv(v[i - 2], x) - (3*n - 9)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n - 24 + (4*n - 12)*x)*v[i - 1] + (4*n - 12)*x*deriv(v[i - 1], x))/(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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.?
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741