A038333 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(40 + 42\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1600 + 3360\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1600	3360	1764								6724
1	64000	201600	211680	74088							551368
2	2560000	10752000	16934400	11854080	3111696						45212176
3	102400000	537600000	1128960000	1185408000	622339200	130691232					3707398432
4	4096000000	25804800000	67737600000	94832640000	74680704000	31365895680	5489031744				304006671424
5	163840000000	1204224000000	3793305600000	6638284800000	6970199040000	4391225395200	1536928888320	230539333248			24928547056768
6	6553600000000	55050240000000	202309632000000	424850227200000	557615923200000	468397375488000	245908622131200	73772586639360	9682651996416		2044140858654976
7	262144000000000	2477260800000000	10404495360000000	25491013632000000	40148346470400000	42155763793920000	29509034655744000	13279065595084800	3485754718709760	406671383849472	167619550409708032

All 15 rows

Technical details

Node id: n-509590a43fbc65c5 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-640a8dc8403e6c62

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:               1600,                3360,                 1764;
  1:              64000,              201600,               211680,                74088;
  2:            2560000,            10752000,             16934400,             11854080,               3111696;
  3:          102400000,           537600000,           1128960000,           1185408000,             622339200,             130691232;
  4:         4096000000,         25804800000,          67737600000,          94832640000,           74680704000,           31365895680,            5489031744;
  5:       163840000000,       1204224000000,        3793305600000,        6638284800000,         6970199040000,         4391225395200,         1536928888320,          230539333248;
  6:      6553600000000,      55050240000000,      202309632000000,      424850227200000,       557615923200000,       468397375488000,       245908622131200,        73772586639360,         9682651996416;
  7:    262144000000000,    2477260800000000,    10404495360000000,    25491013632000000,     40148346470400000,     42155763793920000,     29509034655744000,     13279065595084800,      3485754718709760,      406671383849472;
  8:  10485760000000000,  110100480000000000,   520224768000000000,  1456629350400000000,   2676556431360000000,   3372461103513600000,   2950903465574400000,   1770542079344640000,    697150943741952000,   162668553539788800,   17080198121677824;
  9: 419430400000000000, 4844421120000000000, 25433210880000000000, 80114614272000000000, 168240689971200000000, 247313814257664000000, 259679504970547200000, 194759628727910400000, 102248805082152960000, 35787081778753536000, 7515287173538242560, 717368321110468608;

%F Formula

The row polynomials satisfy P(n,x) = (40+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1600 + 3360*x + 1764*x^2.

%t Mathematica

P[0, x_] := 1600 + 3360*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(40 + 42*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] = 1600 + 3360*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (40 + 42*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)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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

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.