A144698 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(-4 - 6\,x - 4\,x^{2} - 6\,x^{3} - 4\,x^{4} - 2\,n - 5\,n x - 9\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 2\,x^{2} + 2\,x^{4} + 2\,x^{5} - n x - 2\,n x^{2} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 9\,x + 6\,x^{2} + 2\,x^{3} + 3\,n + 5\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	14	45	16													75
3	33	258	305	64												660
4	72	1133	2994	1845	256											6300
5	151	4330	21500	28010	10505	1024										65520
6	310	15249	129000	303490	231090	57645	4096									740880
7	629	51010	690591	2670980	3576715	1756986	308705	16384								9072000
8	1268	164973	3423610	20457419	44052720	37335907	12623522	1625445	65536							119750400
9	2547	521538	16082114	142092790	462940716	621616478	357424102	87066666	8454905	262144						1696464000
10	5106	1623185	72662880	919012440	4335390780	8689276218	7832324640	3208965120	582566610	43585245	1048576					25686460800
11	10225	4997194	319034285	5634675720	37240307970	106645816620	142173241602	90607688040	27434408205	3809339930	223169105	4194304				414096883200
12	20464	15267645	1371032706	33164933105	299351699640	1184850517962	2237002170060	2095028366778	981039171720	225787507385	24471878274	1136817045	16777216			7083236160000
13	40943	46396378	5797005000	189109350754	2284497639155	12177348105204	31492680908664	41518024556820	28440361916757	10080057931570	1803267674528	155061458058	5767971305	67108864		128152088064000
14	81902	140458353	24208494664	1051747919710	16728886228806	117625147789931	405889496408064	728521057560852	699583864735770	361559879385463	99304455396792	14060783965358	971955898034	29175400845	268435456	2445351068160000

Technical details

Node id: n-510f2912b6542c00 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-68412ed67a0586d4

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:    5,      4;
  2:   14,     45,       16;
  3:   33,    258,      305,        64;
  4:   72,   1133,     2994,      1845,       256;
  5:  151,   4330,    21500,     28010,     10505,      1024;
  6:  310,  15249,   129000,    303490,    231090,     57645,      4096;
  7:  629,  51010,   690591,   2670980,   3576715,   1756986,    308705,    16384;
  8: 1268, 164973,  3423610,  20457419,  44052720,  37335907,  12623522,  1625445,   65536;
  9: 2547, 521538, 16082114, 142092790, 462940716, 621616478, 357424102, 87066666, 8454905, 262144;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*x)); 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 of 4-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.