A038321 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 21\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 625 + 1050\,x + 441\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	625	1050	441								2116
1	15625	39375	33075	9261							97336
2	390625	1312500	1653750	926100	194481						4477456
3	9765625	41015625	68906250	57881250	24310125	4084101					205962976
4	244140625	1230468750	2583984375	2894062500	1823259375	612615150	85766121				9474296896
5	6103515625	35888671875	90439453125	126615234375	106356796875	53603825625	15009071175	1801088541			435817657216
6	152587890625	1025390625000	3014648437500	5064609375000	5317839843750	3573588375000	1500907117500	360217708200	37822859361		20047612231936
7	3814697265625	28839111328125	96899414062500	189922851562500	239302792968750	201014346093750	112568033812500	40524492172500	8510143356225	794280046581	922190162669056

All 15 rows

Technical details

Node id: n-2c7754d5ac44f890 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-21bc025a5642ae0d

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:              625,              1050,               441;
  1:            15625,             39375,             33075,               9261;
  2:           390625,           1312500,           1653750,             926100,             194481;
  3:          9765625,          41015625,          68906250,           57881250,           24310125,            4084101;
  4:        244140625,        1230468750,        2583984375,         2894062500,         1823259375,          612615150,           85766121;
  5:       6103515625,       35888671875,       90439453125,       126615234375,       106356796875,        53603825625,        15009071175,         1801088541;
  6:     152587890625,     1025390625000,     3014648437500,      5064609375000,      5317839843750,      3573588375000,      1500907117500,       360217708200,       37822859361;
  7:    3814697265625,    28839111328125,    96899414062500,    189922851562500,    239302792968750,    201014346093750,    112568033812500,     40524492172500,     8510143356225,      794280046581;
  8:   95367431640625,   801086425781250,  3028106689453125,   6782958984375000,   9970949707031250,  10050717304687500,   7035502113281250,   3377041014375000,  1063767919528125,   198570011645250,   16679880978201;
  9: 2384185791015625, 22029876708984375, 92525482177734375, 233164215087890625, 391715881347656250, 460657876464843750, 386952616230468750, 232171569738281250, 97512059290078125, 27303376601221875, 4586967269005275, 350277500542221;

%F Formula

The row polynomials satisfy P(n,x) = (25+21*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 1050*x + 441*x^2.

%t Mathematica

P[0, x_] := 625 + 1050*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 21*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] = 625 + 1050*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 21*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_A007318 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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.