A024462 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 18\,x + 4\,n + 9\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 4 + 12\,x + 9\,x^{2}\)

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	3														5
2	4	12	9													25
3	16	84	144	81												325
4	64	480	1332	1620	729											4225
5	256	2496	9648	18468	17496	6561										54925
6	1024	12288	61056	160704	236196	183708	59049									714025
7	4096	58368	354816	1192320	2391120	2860596	1889568	531441								9282325
8	16384	270336	1944576	7962624	20295360	32962464	33303636	19131876	4782969							120670225
9	65536	1228800	10211328	49351680	152845056	314508096	429876720	376260228	191318760	43046721						1568712925
10	262144	5505024	51904512	289308672	1055545344	2633637888	4550079744	5373931392	4151617092	1894055724	387420489					20393268025
11	1048576	24379392	257163264	1624375296	6825959424	20034459648	41903059968	62446443264	64971850896	44940776724	18596183472	3486784401				265112484325
12	4194304	106954752	1248067584	8811970560	41923215360	141571473408	347922376704	626913312768	821905392960	764509764960	478851724404	181312788852	31381059609			3446462296225
13	16777216	465567744	5954863104	46480490496	247000596480	943594831872	2665832767488	5638954641408	8929841386752	10455187596480	8795994782256	5034916675044	1757339338104	282429536481		44804009850925
14	67108864	2013265920	28009562112	239515729920	1406326800384	5997384695808	19155684556800	46548313473024	86469957319680	122189322866688	129280667497344	99303619740480	52343607427812	16945772188860	2541865828329	582452128062025

Technical details

Node id: n-da3bba3daee1d05e · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bbdfbe7cf94599f2

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     2,       3;
  2:     4,      12,        9;
  3:    16,      84,      144,       81;
  4:    64,     480,     1332,     1620,       729;
  5:   256,    2496,     9648,    18468,     17496,      6561;
  6:  1024,   12288,    61056,   160704,    236196,    183708,     59049;
  7:  4096,   58368,   354816,  1192320,   2391120,   2860596,   1889568,    531441;
  8: 16384,  270336,  1944576,  7962624,  20295360,  32962464,  33303636,  19131876,   4782969;
  9: 65536, 1228800, 10211328, 49351680, 152845056, 314508096, 429876720, 376260228, 191318760, 43046721;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(9*n-18)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 4 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (9*n - 18)*x)*P[n - 1, x])/(n - 2)]];
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 + 3*x; v[3] = 4 + 12*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (9*n - 18)*x)*v[i - 1])/(n - 2)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows, arising in enumeration of catafusenes.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.