A038284 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 676 + 1248\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	676	1248	576															2500
1	17576	48672	44928	13824														125000
2	456976	1687296	2336256	1437696	331776													6250000
3	11881376	54837120	101237760	93450240	43130880	7962624												312500000
4	308915776	1710918144	3948272640	4859412480	3364208640	1242169344	191102976											15625000000
5	8031810176	51897850368	143717124096	221103267840	204095324160	113037410304	34780741632	4586471424										781250000000
6	208827064576	1542107553792	4982193635328	9197895942144	10612956856320	7837260447744	3617197129728	953986056192	110075314176									39062500000000
7	5429503678976	45106645948416	166547615809536	358717941743616	496686380875776	458479736193024	282141376118784	111616368574464	25757623517184	2641807540224								1953125000000000
8	141167095653376	1303080882954240	5412797513809920	13323809264762880	21523076504616960	23840946282037248	18339189447720960	9673418609786880	3348491057233920	686869960458240	63403380965376							97656250000000000
9	3670344486987776	37268113252491264	172006676549959680	476326181215272960	879371411474350080	1136418439443775488	1049001636409638912	691649430599761920	319222814122967040	98222404345528320	18133366956097536	1521681143169024						4882812500000000000
10	95428956661682176	1057059212252479488	5366608308358742016	16512640948796129280	34295485047499653120	50651793300922564608	54548085093301223424	43158924469425143808	24899379501591429120	10215130051934945280	2828805245151215616	474764516668735488	36520347436056576					244140625000000000000
11	2481152873203736576	29773834478444838912	164901237111386800128	558127264069309169664	1287985994006098083840	2140038266963978354688	2633893251647973359616	2431286078444283101184	1683198054307580608512	863178489388502876160	318712057620370292736	80235203317016297472	12343877433387122688	876488338465357824				12207031250000000000000
12	64509974703297150976	833667365396455489536	5002004192378732937216	18468938556475321614336	46882690181821970251776	86552658797209791234048	119842142949982787862528	126426876079102721261568	102114015294659890249728	62839394027483009384448	29002797243453696638976	9735204669131310759936	2246585692876456329216	319041755201390247936	21035720123168587776			610351562500000000000000
13	1677259342285725925376	23223590893186974351360	150060125771361988116480	600240503085447952465920	1662204470082778945290240	3375553693091181858127872	5193159527832587474042880	6163310208856257661501440	5689209423559622456770560	4084560611786395609989120	2262218184989388337840128	949182455240302799093760	292056140073939322798080	62213142264271098347520	8203930848035749232640	504857282956046106624		30517578125000000000000000
14	43608742899428874059776	644067587437718755344384	4458929451491899075461120	19207696098734334478909440	57623088296203003436728320	127657303302357422998290432	216035436357835638920183808	284881894098244798576066560	295838890025100367752069120	242739602071877224822210560	156847127492597591423582208	78971980275993192884600832	30373838567689689571000320	8626889060645592304189440	1706417616391435840389120	210020629709715180355584	12116574790945106558976	1525878906250000000000000000

Technical details

Node id: n-5327121044655cf2 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-42b2d489dcb16cdd

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:              676,              1248,                576;
  1:            17576,             48672,              44928,              13824;
  2:           456976,           1687296,            2336256,            1437696,             331776;
  3:         11881376,          54837120,          101237760,           93450240,           43130880,             7962624;
  4:        308915776,        1710918144,         3948272640,         4859412480,         3364208640,          1242169344,           191102976;
  5:       8031810176,       51897850368,       143717124096,       221103267840,       204095324160,        113037410304,         34780741632,         4586471424;
  6:     208827064576,     1542107553792,      4982193635328,      9197895942144,     10612956856320,       7837260447744,       3617197129728,       953986056192,       110075314176;
  7:    5429503678976,    45106645948416,    166547615809536,    358717941743616,    496686380875776,     458479736193024,     282141376118784,    111616368574464,     25757623517184,     2641807540224;
  8:  141167095653376,  1303080882954240,   5412797513809920,  13323809264762880,  21523076504616960,   23840946282037248,   18339189447720960,   9673418609786880,   3348491057233920,   686869960458240,    63403380965376;
  9: 3670344486987776, 37268113252491264, 172006676549959680, 476326181215272960, 879371411474350080, 1136418439443775488, 1049001636409638912, 691649430599761920, 319222814122967040, 98222404345528320, 18133366956097536, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (26+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1248*x + 576*x^2.

%t Mathematica

P[0, x_] := 676 + 1248*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 24*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] = 676 + 1248*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 24*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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.