A038328 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 26\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 144 + 336\,x + 196\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	336	196								676
1	1728	6048	7056	2744							17576
2	20736	96768	169344	131712	38416						456976
3	248832	1451520	3386880	3951360	2304960	537824					11881376
4	2985984	20901888	60963840	94832640	82978560	38723328	7529536				308915776
5	35831808	292626432	1024192512	1991485440	2323399680	1626379776	632481024	105413504			8031810176
6	429981696	4013162496	16387080192	38236520448	55761592320	52044152832	30359089152	10119696384	1475789056		208827064576
7	5159780352	54177693696	252829237248	688257368064	1204450394112	1405192126464	1092927209472	546463604736	159385218048	20661046784	5429503678976

All 15 rows

Technical details

Node id: n-0649e5ac47163e39 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8e9037e0ccb64a62

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:          144,           336,            196;
  1:         1728,          6048,           7056,            2744;
  2:        20736,         96768,         169344,          131712,           38416;
  3:       248832,       1451520,        3386880,         3951360,         2304960,          537824;
  4:      2985984,      20901888,       60963840,        94832640,        82978560,        38723328,         7529536;
  5:     35831808,     292626432,     1024192512,      1991485440,      2323399680,      1626379776,       632481024,       105413504;
  6:    429981696,    4013162496,    16387080192,     38236520448,     55761592320,     52044152832,     30359089152,     10119696384,      1475789056;
  7:   5159780352,   54177693696,   252829237248,    688257368064,   1204450394112,   1405192126464,   1092927209472,    546463604736,    159385218048,     20661046784;
  8:  61917364224,  722369249280,  3792438558720,  11798697738240,  24089007882240,  33724611035136,  32787816284160,  21858544189440,   9563113082880,   2479325614080,   289254654976;
  9: 743008370688, 9535274090496, 55622432194560, 194678512680960, 454249862922240, 741941442772992, 865598349901824, 721331958251520, 420776975646720, 163635490529280, 38181614456832, 4049565169664;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (12+26*x+14*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 336*x + 196*x^2.

%t Mathematica

P[0, x_] := 144 + 336*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 26*x + 14*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 336*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 26*x + 14*x^2)*v[i - 1])/(1 + x)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - 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
  - Parent-row provers of polytool polytool ? 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 ?

Candidate equalities ?

A038284 → basis A013609 → reverse 15 rows; no all-index equality certificate was found

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.