A038763 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

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	4														6
2	8	40	48													96
3	32	256	672	576												1536
4	128	1408	5760	10368	6912											24576
5	512	7168	39936	110592	152064	82944										393216
6	2048	34816	245760	921600	1935360	2156544	995328									6291456
7	8192	163840	1400832	6635520	18800640	31850496	29859840	11943936								100663296
8	32768	753664	7569408	43352064	154828800	353009664	501645312	406093824	143327232							1610612736
9	131072	3407872	39321600	264241152	1139539968	3269984256	6242697216	7644119040	5446434816	1719926784						25769803776
10	524288	15204352	198180864	1528823808	7729053696	26754416640	64210599936	105488842752	113515167744	72236924928	20639121408					412316860416
11	2097152	67108864	975175680	8493465600	49262100480	199766310912	577895399424	1192482570240	1719926784000	1651129712640	949399584768	247669456896				6597069766656
12	8388608	293601280	4706009088	45675970560	298969989120	1390210449408	4708777328640	11704675074048	21189497978880	27243640258560	23611154890752	12383472844800	2972033482752			105553116266496
13	33554432	1275068416	22347251712	239175991296	1743991603200	9148481667072	35517634707456	103324028239872	225214092804096	363248536780800	421368302665728	332867750068224	160489808068608	35664401793024		1688849860263936
14	134217728	5502926848	104689827840	1224870985728	9846078308352	57521825906688	251852318834688	839507729448960	2140744710094848	4155563260772352	6044455652032512	6387890632261632	4636372233093120	2068535303995392	427972821516288	27021597764222976

Technical details

Node id: n-4636bdf7e3e77f22 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-6d5781c4134cee14

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:      8,      40,       48;
  3:     32,     256,      672,       576;
  4:    128,    1408,     5760,     10368,       6912;
  5:    512,    7168,    39936,    110592,     152064,      82944;
  6:   2048,   34816,   245760,    921600,    1935360,    2156544,     995328;
  7:   8192,  163840,  1400832,   6635520,   18800640,   31850496,   29859840,   11943936;
  8:  32768,  753664,  7569408,  43352064,  154828800,  353009664,  501645312,  406093824,  143327232;
  9: 131072, 3407872, 39321600, 264241152, 1139539968, 3269984256, 6242697216, 7644119040, 5446434816, 1719926784;

%F Formula

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

%t Mathematica

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