A038763 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

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	8	20	12													40
3	32	128	168	72												400
4	128	704	1440	1296	432											4000
5	512	3584	9984	13824	9504	2592										40000
6	2048	17408	61440	115200	120960	67392	15552									400000
7	8192	81920	350208	829440	1175040	995328	466560	93312								4000000
8	32768	376832	1892352	5419008	9676800	11031552	7838208	3172608	559872							40000000
9	131072	1703936	9830400	33030144	71221248	102187008	97542144	59719680	21275136	3359232						400000000
10	524288	7602176	49545216	191102976	483065856	836075520	1003290624	824131584	443418624	141087744	20155392					4000000000
11	2097152	33554432	243793920	1061683200	3078881280	6242697216	9029615616	9316270080	6718464000	3224862720	927148032	120932352				40000000000
12	8388608	146800640	1176502272	5709496320	18685624320	43444076544	73574645760	91442774016	82771476480	53210234880	23057768448	6046617600	725594112			400000000000
13	33554432	637534208	5586812928	29896998912	108999475200	285890052096	554963042304	807218970624	879742550016	709469798400	411492483072	162533081088	39182082048	4353564672		4000000000000
14	134217728	2751463424	26172456960	153108873216	615379894272	1797557059584	3935192481792	6558654136320	8362284023808	8116334493696	5902788722688	3119087222784	1131926814720	252506750976	26121388032	40000000000000

Technical details

Node id: n-079690ff502c56c6 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b821bbf6b18e27b2

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:      8,      20,      12;
  3:     32,     128,     168,       72;
  4:    128,     704,    1440,     1296,      432;
  5:    512,    3584,    9984,    13824,     9504,      2592;
  6:   2048,   17408,   61440,   115200,   120960,     67392,    15552;
  7:   8192,   81920,  350208,   829440,  1175040,    995328,   466560,    93312;
  8:  32768,  376832, 1892352,  5419008,  9676800,  11031552,  7838208,  3172608,   559872;
  9: 131072, 1703936, 9830400, 33030144, 71221248, 102187008, 97542144, 59719680, 21275136, 3359232;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (6*n - 6)*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] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (6*n - 6)*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 Triangular matrix arising in enumeration of catafusenes, read by rows.?
  - 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.