A024462 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 12\,x + 4\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 4 + 8\,x + 4\,x^{2}\)

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	2														4
2	4	8	4													16
3	16	56	64	24												160
4	64	320	592	480	144											1600
5	256	1664	4288	5472	3456	864										16000
6	1024	8192	27136	47616	46656	24192	5184									160000
7	4096	38912	157696	353280	472320	376704	165888	31104								1600000
8	16384	180224	864256	2359296	4008960	4340736	2923776	1119744	186624							16000000
9	65536	819200	4538368	14622720	30191616	41416704	37739520	22021632	7464960	1119744						160000000
10	262144	3670016	23068672	85721088	208502784	346816512	399458304	314523648	161989632	49268736	6718464					1600000000
11	1048576	16252928	114294784	481296384	1348337664	2638282752	3678732288	3654844416	2535100416	1169012736	322486272	40310784				16000000000
12	4194304	71303168	554696704	2610954240	8281128960	18643156992	30544625664	36691771392	32069468160	19886653440	8304021504	2096160768	241864704			160000000000
13	16777216	310378496	2646605824	13771997184	48790241280	124259401728	234037444608	330034839552	348428500992	271963422720	152536006656	58208772096	13544423424	1451188224		1600000000000
14	67108864	1342177280	12448694272	70967623680	277792948224	789779054592	1681706188800	2724364025856	3373923041280	3178424696832	2241924562944	1148051128320	403430326272	87071293440	8707129344	16000000000000

Technical details

Node id: n-080bedf92b7d54b8 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-f26e13ac80424634

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,      2;
  2:     4,      8,       4;
  3:    16,     56,      64,       24;
  4:    64,    320,     592,      480,      144;
  5:   256,   1664,    4288,     5472,     3456,      864;
  6:  1024,   8192,   27136,    47616,    46656,    24192,     5184;
  7:  4096,  38912,  157696,   353280,   472320,   376704,   165888,    31104;
  8: 16384, 180224,  864256,  2359296,  4008960,  4340736,  2923776,  1119744,  186624;
  9: 65536, 819200, 4538368, 14622720, 30191616, 41416704, 37739520, 22021632, 7464960, 1119744;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(6*n-12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 4 + 8*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 4 + 8*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (6*n - 12)*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 + 2*x; v[3] = 4 + 8*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (6*n - 12)*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 ?

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.