A225117 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 3\,n\right)\,P_{n}(x) = \left(50 - 220\,x + 440\,x^{2} - 30\,n + 192\,n x - 549\,n x^{2} - 36\,n^{2} x + 216\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 150\,x^{2} + 120\,x^{3} - 18\,n x + 135\,n x^{2} - 117\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-50 + 128\,x + 21\,n - 96\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 24\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	13	2													15
4	93	120	3												216
5	545	2262	753	4											3564
6	2933	28996	32034	4072	5										68040
7	15177	308226	787314	355992	20445	6									1487160
8	77101	2951472	14670861	15481408	3462495	98256	7								36741600
9	388321	26541614	232214799	471034636	252380015	31050078	458689	8							1014068160
10	1948005	229419036	3310798788	11548400616	11944773090	3649561740	264125484	2097072	9						30951123840
11	9754361	1932752538	43989614856	245551177920	434029154850	259329695004	48653106432	2167527744	9437085	10					1035672220800
12	48803549	16008264520	556363983591	4728078526656	13210336108050	13519953673200	5052188488830	612391223424	17342956449	41942920	11				37712753971200
13	244087377	131091936198	6792350929341	84738565492884	354597873319890	571594736727540	368572803809490	90983257928784	7392347000541	136231834710	184549233	12			1484939687616000
14	1220588437	1065333431220	80811635255958	1439921018262440	8676244018090755	20793278590195896	21147524485078740	9105181690093344	1545008210224155	86496607605620	1055444350422	805306200	13		62876589058483200
15	6103269865	8612990994098	943259992807758	23499108512920424	197893383946728235	676152892346640570	1020087213638505684	695938112761279584	208698508518180687	25084676702173190	988403235926590	8090358493368	3489660733	14	2849302272597580800

Technical details

Node id: n-8eee865029988ec1 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5c9708a73b43501c

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       0;
   2:       1;
   3:      13,         2;
   4:      93,       120,          3;
   5:     545,      2262,        753,           4;
   6:    2933,     28996,      32034,        4072,           5;
   7:   15177,    308226,     787314,      355992,       20445,          6;
   8:   77101,   2951472,   14670861,    15481408,     3462495,      98256,         7;
   9:  388321,  26541614,  232214799,   471034636,   252380015,   31050078,    458689,       8;
  10: 1948005, 229419036, 3310798788, 11548400616, 11944773090, 3649561740, 264125484, 2097072, 9;

%F Formula

The row polynomials satisfy (3*n-8)*P(n,x) = -(30*n-50+(36*n^2-192*n+220)*x+(27*n^3-216*n^2+549*n-440)*x^2)*P(n-2,x) - ((18*n-30)*x+(27*n^2-135*n+150)*x^2-(27*n^2-117*n+120)*x^3)*d/dx P(n-2,x) + (21*n-50+(18*n^2-96*n+128)*x)*P(n-1,x) + ((9*n-24)*x-(9*n-24)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*P[n - 2, x] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*D[P[n - 2, x], x] + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*P[n - 1, x] + ((9*n - 24)*x - (9*n - 24)*x^2)*D[P[n - 1, x], x])/(3*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*v[i - 2] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*deriv(v[i - 2], x) + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*v[i - 1] + ((9*n - 24)*x - (9*n - 24)*x^2)*deriv(v[i - 1], x))/(3*n - 8)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.?
  - 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.