A095660 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 3\)

\(P_{1}(x) = 4 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	3															3
1	4	12														16
2	8	40	48													96
3	16	112	256	192												576
4	32	288	960	1408	768											3456
5	64	704	3072	6656	7168	3072										20736
6	128	1664	8960	25600	40960	34816	12288									124416
7	256	3840	24576	87040	184320	233472	163840	49152								746496
8	512	8704	64512	272384	716800	1204224	1261568	753664	196608							4478976
9	1024	19456	163840	802816	2523136	5275648	7340032	6553600	3407872	786432						26873856
10	2048	43008	405504	2260992	8257536	20643840	35782656	42467328	33030144	15204352	3145728					161243136
11	4096	94208	983040	6144000	25559040	74317824	154140672	228065280	235929600	162529280	67108864	12582912				967458816
12	8192	204800	2342912	16220160	75694080	250871808	605552640	1072693248	1384120320	1268776960	784334848	293601280	50331648			5804752896
13	16384	442368	5505024	41811968	216268800	804519936	2214592512	4567597056	7059013632	8074035200	6643777536	3724541952	1275068416	201326592		34828517376
14	32768	950272	12779520	105644032	599785472	2474115072	7647264768	17993564160	32388415488	44384124928	45583695872	34024194048	17448304640	5502926848	805306368	208971104256

Technical details

Node id: n-301af6d19610a18d · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-1c35c8e273da035a

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:    3;
  1:    4,    12;
  2:    8,    40,     48;
  3:   16,   112,    256,    192;
  4:   32,   288,    960,   1408,     768;
  5:   64,   704,   3072,   6656,    7168,    3072;
  6:  128,  1664,   8960,  25600,   40960,   34816,   12288;
  7:  256,  3840,  24576,  87040,  184320,  233472,  163840,   49152;
  8:  512,  8704,  64512, 272384,  716800, 1204224, 1261568,  753664,  196608;
  9: 1024, 19456, 163840, 802816, 2523136, 5275648, 7340032, 6553600, 3407872, 786432;

%F Formula

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

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (4*n - 4)*x)*P[n - 1, 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] = 3; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (4*n - 4)*x)*v[i - 1])/(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 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Pascal (1,3) triangle.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - 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.