A193729 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 12\,x + 7\,n + 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	21	78	72													171
3	147	798	1440	864												3249
4	1029	7350	19656	23328	10368											61731
5	7203	63798	225792	399168	352512	124416										1172889
6	50421	533022	2346120	5503680	7257600	5101056	1492992									22284891
7	352947	4336206	22819104	66679200	116847360	122798592	71663616	17915904								423412929
8	2470629	34588806	211768200	740583648	1618081920	2261758464	1975228416	985374720	214990848							8044845651
9	17294403	271769190	1897443072	7725303936	20213577216	35249292288	40967700480	30600364032	13329432576	2579890176						152852067369
10	121060821	2109917166	16543331784	76846444416	234198687744	489307972608	709765410816	705814953984	460510396416	178012422144	30958682112					2904189280011
11	847425747	16222150014	141122328480	736445092320	2561548147200	6235540061184	10840053547008	13457889607680	11693352222720	6772211712000	2352859840512	371504185344				55179596320209
12	5931980229	123724159062	1182522099528	6848583588000	26768178138240	74387358194688	150706855563264	224285869817856	243348140851200	187725708656640	97736559427584	30834847383552	4458050224128			1048412330083971
13	41523861603	937252876182	9762344605440	62130350310336	269560250023680	841929645021696	1947596287279104	3378483355484160	4394867423772672	4234257650810880	2936864419872768	1388682644815872	401224520171520	53496602689536		19919834271595449
14	290667031221	7059056472510	79583446752264	552060587437632	2632485953889792	9128230515436032	23736329751214080	47020538935738368	71305872232218624	82378212640948224	71369142748839936	44963151552184320	19472763378991104	5189170460884992	641959232274432	378476851160313531

Technical details

Node id: n-b0895ced9a2cb6fd · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bdb6dddf14699d20

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:        3,         6;
  2:       21,        78,         72;
  3:      147,       798,       1440,        864;
  4:     1029,      7350,      19656,      23328,       10368;
  5:     7203,     63798,     225792,     399168,      352512,      124416;
  6:    50421,    533022,    2346120,    5503680,     7257600,     5101056,     1492992;
  7:   352947,   4336206,   22819104,   66679200,   116847360,   122798592,    71663616,    17915904;
  8:  2470629,  34588806,  211768200,  740583648,  1618081920,  2261758464,  1975228416,   985374720,   214990848;
  9: 17294403, 271769190, 1897443072, 7725303936, 20213577216, 35249292288, 40967700480, 30600364032, 13329432576, 2579890176;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(12*n-12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (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] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (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 Mirror of the triangle A193728.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.