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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	95367431640625	801086425781250	3028106689453125	6782958984375000	9970949707031250	10050717304687500	7035502113281250	3377041014375000	1063767919528125	198570011645250	16679880978201							42420747482776576
9	2384185791015625	22029876708984375	92525482177734375	233164215087890625	391715881347656250	460657876464843750	386952616230468750	232171569738281250	97512059290078125	27303376601221875	4586967269005275	350277500542221						1951354384207722496
10	59604644775390625	600814819335937500	2775764465332031250	7772140502929687500	14689345550537109375	19742480419921875000	19347630811523437500	13930294184296875000	7313404446755859375	2730337660122187500	688045090350791250	105083250162666300	7355827511386641					89762301673555234816
11	1490116119384765625	16272068023681640625	82011222839355468750	252594566345214843750	530448589324951171875	802038267059326171875	898282859106445312500	754557601649414062500	475371289039130859375	221839934884927734375	74538218121335718750	17076028151433273750	2390643941200658325	154472377739119461				4129065876983540801536
12	37252902984619140625	438094139099121093750	2391993999481201171875	8037099838256835937500	18565700626373291015625	31190377052307128906250	39299875085906982421875	37727880082470703125000	27729991860615966796875	15528795441944941406250	6522094085616875390625	1992203284333881937500	418362689710115206875	54065332208691811350	3243919932521508681			189937030341242876870656
13	931322574615478515625	11734664440155029296875	68999826908111572265625	251159369945526123046875	632921612262725830078125	1169639139461517333984375	1637494795246124267578125	1768494378865814208984375	1485535278247283935546875	970549715121558837890625	489157056421265654296875	186769057906301431640625	52295336213764400859375	10137249789129714628125	1216469974695565755375	68122318582951682301		8737103395697172336050176
14	23283064365386962890625	312924385070800781250000	1971423625946044921875000	7727980613708496093750000	21097387075424194335937500	42532332344055175781250000	65499791809844970703125000	78599750171813964843750000	74276763912364196777343750	55459983721231933593750000	32610470428084376953125000	14941524632504114531250000	5229533621376440085937500	1351633305217295283750000	243293994939113151075000	27248927433180672920400	1430568690241985328321	401906756202069927458308096

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.