A109983 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n - n x - 3\,n x^{2} + n x^{3} - 2\,n^{2} + 2\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-4\,x^{2} - 6\,x^{3} + 2\,x^{5} - 6\,n x - 8\,n x^{2} + 2\,n x^{3} + 3\,n x^{4} + 6\,n^{2} x + 6\,n^{2} x^{2} - 2\,n^{2} x^{3} - n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x^{3} - 2\,x^{4} + 2\,x^{5} + 2\,x^{6} - 3\,n x^{2} - 3\,n x^{3} + 4\,n x^{4} + 4\,n x^{5} + 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	5	2																											8
2	0	2	19	30	6																									57
3	0	0	3	49	162	150	20																							384
4	0	0	0	4	101	560	1070	700	70																					2505
5	0	0	0	0	5	181	1500	4690	6230	3150	252																			16008
6	0	0	0	0	0	6	295	3402	15540	33180	33642	13860	924																	100849
7	0	0	0	0	0	0	7	449	6860	42756	131250	211134	172788	60060	3432															628736
8	0	0	0	0	0	0	0	8	649	12672	102900	425040	971586	1249248	856284	257400	12870													3888657
9	0	0	0	0	0	0	0	0	9	901	21870	223740	1189650	3621618	6558552	7009860	4131270	1093950	48620											23900040
10	0	0	0	0	0	0	0	0	0	10	1211	35750	449130	2977260	11561550	27663636	41355600	37777740	19520930	4618900	184756									146146473
11	0	0	0	0	0	0	0	0	0	0	11	1585	55902	845130	6816810	32738706	99219120	194752272	247451490	197227030	90715196	19399380	705432							889928064
12	0	0	0	0	0	0	0	0	0	0	0	12	2029	84240	1507506	14514500	84216132	313657344	775391760	1287010296	1420311750	1003594592	415852164	81124680	2704156					5399971161
13	0	0	0	0	0	0	0	0	0	0	0	0	13	2549	123032	2570750	29089060	200187988	895638744	2707044912	5632748550	8086308230	7881321448	4999984444	1884796732	338019500	10400600			32668236552
14	0	0	0	0	0	0	0	0	0	0	0	0	0	14	3151	174930	4218760	55382600	445368924	2351204856	8496928440	21524997780	38588600050	48753228524	42525557256	24472611800	8460888100	1404081000	40116600	197123362785

Technical details

Node id: n-45f4a43cf1c73c51 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-97a44f7e6824c858

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1, 5,  2;
  2: 0, 2, 19, 30,   6;
  3: 0, 0,  3, 49, 162, 150,   20;
  4: 0, 0,  0,  4, 101, 560, 1070,  700,    70;
  5: 0, 0,  0,  0,   5, 181, 1500, 4690,  6230,  3150,    252;
  6: 0, 0,  0,  0,   0,   6,  295, 3402, 15540, 33180,  33642,  13860,     924;
  7: 0, 0,  0,  0,   0,   0,    7,  449,  6860, 42756, 131250, 211134,  172788,   60060,    3432;
  8: 0, 0,  0,  0,   0,   0,    0,    8,   649, 12672, 102900, 425040,  971586, 1249248,  856284,  257400,   12870;
  9: 0, 0,  0,  0,   0,   0,    0,    0,     9,   901,  21870, 223740, 1189650, 3621618, 6558552, 7009860, 4131270, 1093950, 48620;

%F Formula

The row polynomials satisfy (n^3-2*n^2+n-n*x+(2*n^2-3*n)*x^2+n*x^3)*P(n,x) = -((n^3-6*n^2+6*n)*x-(6*n^2-8*n-4)*x^2+(2*n^2-2*n+6)*x^3-3*n*x^4-2*x^5)*P(n-1,x) + ((2*n^2-3*n)*x^2+(2*n^2-3*n-2)*x^3+(4*n-2)*x^4+(4*n+2)*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 6*n)*x - (6*n^2 - 8*n - 4)*x^2 + (2*n^2 - 2*n + 6)*x^3 - 3*n*x^4 - 2*x^5)*P[n - 1, x] + ((2*n^2 - 3*n)*x^2 + (2*n^2 - 3*n - 2)*x^3 + (4*n - 2)*x^4 + (4*n + 2)*x^5 + 2*x^6)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n - n*x + (2*n^2 - 3*n)*x^2 + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-((n^3 - 6*n^2 + 6*n)*x - (6*n^2 - 8*n - 4)*x^2 + (2*n^2 - 2*n + 6)*x^3 - 3*n*x^4 - 2*x^5)*v[i - 1] + ((2*n^2 - 3*n)*x^2 + (2*n^2 - 3*n - 2)*x^3 + (4*n - 2)*x^4 + (4*n + 2)*x^5 + 2*x^6)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n - n*x + (2*n^2 - 3*n)*x^2 + n*x^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 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.