A156365 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-4 - 2\,n - 8\,n x - 8\,n x^{2} - 4\,n^{2} x - 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 8\,x^{3} - n x - 4\,n x^{2} + 12\,n x^{3} - 2\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 4\,x + 3\,n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} + n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 8 + 8\,x\)

\(P_{1}(x) = 22 + 88\,x + 24\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	8	8															16
1	22	88	24														134
2	52	528	624	64													1268
3	114	2416	7248	3648	160												13586
4	240	9528	57984	76224	19200	384											163560
5	494	34344	374856	999616	686880	94848	896										2191934
6	1004	116864	2117616	9996160	14117440	5609472	449792	2048									32410396
7	2026	382720	10924608	83862656	209656640	174793728	42864640	2074624	4608								524566250
8	4072	1221096	52883712	623239296	2515879680	3739435776	1974325248	312600576	9381888	10240							9228981584
9	8166	3826168	244504440	4244382336	26001965760	62404717824	59421352704	20864378880	2203872768	41809920	22528						175430841494
10	16356	11837808	1092802800	27090018240	240899441280	873666049536	1349036871168	866880583680	209818137600	15152394240	184233984	49152					3583832435844
11	32738	36298512	4761858864	164591499200	2054922058080	10740931619328	25062173778432	26303002343424	11850587942400	2031726448640	102216609792	804569088	106496				78315755164994
12	65504	110597912	20354181888	962638689344	16444319828480	119572837789824	400994780454912	637721801545728	473596411060224	154022190295040	19105792065536	679513571328	3487956992	229376			1823124238332088
13	131038	335461960	85730046120	5464527267136	125138482927840	1231673540412288	5718987837225600	13071972199372800	14780082484947456	8008862907381760	1923513598031872	175575134453760	4465669611520	15028486144	491520		45045837476248814
14	262108	1014248160	357009586800	30294611267840	914766715581120	11925370732180992	74490951974091520	235295499588710400	383096324438184960	318009886191493120	128799153553821696	23266261453701120	1584170539827200	29080523243520	64415662080	1048576	1177442182762911212

Technical details

Node id: n-3e41ebbd7b30e046 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f911b27c2e10a872

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:    8,       8;
  1:   22,      88,        24;
  2:   52,     528,       624,         64;
  3:  114,    2416,      7248,       3648,         160;
  4:  240,    9528,     57984,      76224,       19200,         384;
  5:  494,   34344,    374856,     999616,      686880,       94848,         896;
  6: 1004,  116864,   2117616,    9996160,    14117440,     5609472,      449792,        2048;
  7: 2026,  382720,  10924608,   83862656,   209656640,   174793728,    42864640,     2074624,       4608;
  8: 4072, 1221096,  52883712,  623239296,  2515879680,  3739435776,  1974325248,   312600576,    9381888,    10240;
  9: 8166, 3826168, 244504440, 4244382336, 26001965760, 62404717824, 59421352704, 20864378880, 2203872768, 41809920, 22528;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -(2*n+4+(4*n^2+8*n)*x+(4*n^3+12*n^2+8*n)*x^2)*P(n-2,x) - ((n+2)*x+(2*n^2+4*n)*x^2-(4*n^2+12*n+8)*x^3)*d/dx P(n-2,x) + (3*n+4+(4*n^2+8*n+4)*x)*P(n-1,x) + ((n+1)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 8 + 8*x, P(1,x) = 22 + 88*x + 24*x^2.

%t Mathematica

P[0, x_] := 8 + 8*x;
P[1, x_] := 22 + 88*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (4*n^2 + 8*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (2*n^2 + 4*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((n + 1)*x - (2*n + 2)*x^2)*D[P[n - 1, x], 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] = 8 + 8*x; v[2] = 22 + 88*x + 24*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (4*n^2 + 8*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*v[i - 2] - ((n + 2)*x + (2*n^2 + 4*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((n + 1)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A142075 → D

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.