A024462 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 24\,x + 4\,n + 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 4 + 16\,x + 16\,x^{2}\)

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	4														6
2	4	16	16													36
3	16	112	256	192												576
4	64	640	2368	3840	2304											9216
5	256	3328	17152	43776	55296	27648										147456
6	1024	16384	108544	380928	746496	774144	331776									2359296
7	4096	77824	630784	2826240	7557120	12054528	10616832	3981312								37748736
8	16384	360448	3457024	18874368	64143360	138903552	187121664	143327232	47775744							603979776
9	65536	1638400	18153472	116981760	483065856	1325334528	2415329280	2818768896	1911029760	573308928						9663676416
10	262144	7340032	92274688	685768704	3336044544	11098128384	25565331456	40259026944	41469345792	25225592832	6879707136					154618822656
11	1048576	32505856	457179136	3850371072	21573402624	84425048064	235438866432	467820085248	648985706496	598534520832	330225942528	82556485632				2473901162496
12	4194304	142606336	2218786816	20887633920	132498063360	596581023744	1954856042496	4696546738176	8209783848960	10181966561280	8503318020096	4292937252864	990677827584			39582418599936
13	16777216	620756992	10586423296	110175977472	780643860480	3976300855296	14978396454912	42244459462656	89197696253952	139245272432640	156196870815744	119211565252608	55477958344704	11888133931008		633318697598976
14	67108864	2684354560	49794777088	567740989440	4444687171584	25272929746944	107629196083200	348718595309568	863724298567680	1627353444777984	2295730752454656	2351208710799360	1652450616410112	713288035860480	142657607172096	10133099161583616

Technical details

Node id: n-373403bc1de177ac · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-8987aa5000ed2f45

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,       4;
  2:     4,      16,       16;
  3:    16,     112,      256,       192;
  4:    64,     640,     2368,      3840,      2304;
  5:   256,    3328,    17152,     43776,     55296,      27648;
  6:  1024,   16384,   108544,    380928,    746496,     774144,     331776;
  7:  4096,   77824,   630784,   2826240,   7557120,   12054528,   10616832,    3981312;
  8: 16384,  360448,  3457024,  18874368,  64143360,  138903552,  187121664,  143327232,   47775744;
  9: 65536, 1638400, 18153472, 116981760, 483065856, 1325334528, 2415329280, 2818768896, 1911029760, 573308928;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(12*n-24)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 4 + 16*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 4 + 16*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (12*n - 24)*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 + 4*x; v[3] = 4 + 16*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (12*n - 24)*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

Related ?

Same fingerprint ?

A159854 → x ↦ x+1 → x ↦ x+1

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.