A038321 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(39 + 63\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1521 + 4914\,x + 3969\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1521	4914	3969								10404
1	59319	287469	464373	250047							1061208
2	2313441	14948388	36221094	39007332	15752961						108243216
3	90224199	728733915	2354371110	3803214870	3071827395	992436543					11040808032
4	3518743761	34104747222	137730709935	296650759860	359403805215	232230151062	62523502209				1126162419264
5	137231006679	1551765998601	7520096762451	20246414360445	32705746274565	31699415619963	17068916103057	3938980639167			114868566764928
6	5352009260481	69164427366216	391045031647452	1263376256091768	2551048209416070	3296739224476152	2662750912076892	1228961959420104	248155780267521		11716593810022656
7	208728361158759	3034589250692727	19608115158322236	73907510981368428	179083584301008114	289288866947782338	311541856712996364	215682823878228252	87102678873899871	15633814156853823	1195092568622310912

All 15 rows

Technical details

Node id: n-20046f09233a3fbb · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-8f86ccbca4a04f09

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:               1521,                4914,                 3969;
  1:              59319,              287469,               464373,                250047;
  2:            2313441,            14948388,             36221094,              39007332,              15752961;
  3:           90224199,           728733915,           2354371110,            3803214870,            3071827395,              992436543;
  4:         3518743761,         34104747222,         137730709935,          296650759860,          359403805215,           232230151062,            62523502209;
  5:       137231006679,       1551765998601,        7520096762451,        20246414360445,        32705746274565,         31699415619963,         17068916103057,          3938980639167;
  6:      5352009260481,      69164427366216,      391045031647452,      1263376256091768,      2551048209416070,       3296739224476152,       2662750912076892,       1228961959420104,        248155780267521;
  7:    208728361158759,    3034589250692727,    19608115158322236,     73907510981368428,    179083584301008114,     289288866947782338,     311541856712996364,     215682823878228252,      87102678873899871,      15633814156853823;
  8:   8140406085191601,  131498867530018170,   955895613968209005,   4117704183247669560,  11640432979565527410,   22564531621927022364,   30375331029517145490,   28038767104169672760,   16985022380410474845,    6097187521172990970,    984930291881790849;
  9: 317475837322472439, 5641301417037779493, 45564357599151295905, 220811886826656280155, 713392249747658751270, 1613364010967782099026, 2606203402332571083042, 3007157771922197403510, 2428858200398697902835, 1307846723291606563065, 422535095217288274221, 62050608388552823487;

%F Formula

The row polynomials satisfy P(n,x) = (39+63*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 4914*x + 3969*x^2.

%t Mathematica

P[0, x_] := 1521 + 4914*x + 3969*x^2;
P[n_, x_] := P[n, x] = Expand[(39 + 63*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1521 + 4914*x + 3969*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 63*x)*v[i - 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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

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.