A154233 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(x^{2} - n x^{2} - 5\,n^{2} x^{2} + n^{3} x^{2} + 3\,n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-4}(x) + \left(2\,x + 4\,n x + 2\,n x^{2} + 2\,n^{2} x^{2} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(-1 + x^{2} - n - 6\,n x - n x^{2} + 2\,n^{3} x\right)\,P_{n-2}(x) + \left(-2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 8 + 9\,x + x^{2}\)

\(P_{3}(x) = 20 + 57\,x + 22\,x^{2} + x^{3}\)

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	8	9	1													18
3	20	57	22	1												100
4	40	381	288	43	1											753
5	70	1591	3044	1037	74	1										5817
6	112	4969	28028	18249	2984	117	1									54460
7	168	12837	164786	264085	84492	7337	174	1								533880
8	240	29005	709216	3077519	1869532	317909	16056	247	1							6019725
9	330	59299	2458912	23385013	33862806	10502655	1016204	32117	338	1						71317675
10	440	112169	7276988	129203877	475892184	279033849	48548496	2853961	59816	449	1					942982230
11	572	199377	19075726	567090357	4439569904	6019130145	1841214338	190794757	7220068	105113	582	1				13084400940
12	728	336765	45419776	2093516019	30086829820	98851846529	56883844208	10070308559	655363296	16764109	176016	739	1			198704406565
13	910	545103	100003740	6751630941	160717920930	1094051781367	1416325488712	431502004753	46924035718	2012035575	36236948	283005	922	1		3158421968625
14	1120	851017	206320796	19530150817	715292073832	8810634670541	26584808222984	15149501490057	2723526271360	190739135993	5620331156	73731337	439496	1133	1	54199933691640

Technical details

Node id: n-67767e52c481da9f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d5053c86d2cadfa2

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:   8,     9,       1;
  3:  20,    57,      22,        1;
  4:  40,   381,     288,       43,        1;
  5:  70,  1591,    3044,     1037,       74,        1;
  6: 112,  4969,   28028,    18249,     2984,      117,       1;
  7: 168, 12837,  164786,   264085,    84492,     7337,     174,     1;
  8: 240, 29005,  709216,  3077519,  1869532,   317909,   16056,   247,   1;
  9: 330, 59299, 2458912, 23385013, 33862806, 10502655, 1016204, 32117, 338, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^5-3*n^4-n^3+5*n^2+n-1)*x^2*P(n-4,x) - ((2*n^3-4*n-2)*x+(2*n^3-2*n^2-2*n)*x^2)*P(n-3,x) - (n+1-(2*n^3-6*n)*x+(n-1)*x^2)*P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 8 + 9*x + x^2, P(3,x) = 20 + 57*x + 22*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 8 + 9*x + x^2;
P[3, x_] := 20 + 57*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*P[n - 4, x] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 3, x] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*P[n - 2, x] + (2*n + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];
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] = 8 + 9*x + x^2; v[4] = 20 + 57*x + 22*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*v[i - 4] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*v[i - 3] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*v[i - 2] + (2*n + (2*n - 2)*x)*v[i - 1])/(n - 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.