A272867 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 24\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	16	24	6																											46
2	256	768	768	288	36																									2116
3	4096	18432	32256	27648	12096	2592	216																							97336
4	65536	393216	983040	1327104	1050624	497664	138240	20736	1296																					4477456
5	1048576	7864320	25559040	47185920	54558720	41140224	20459520	6635520	1347840	155520	7776																			205962976
6	16777216	150994944	603979776	1415577600	2158755840	2250768384	1642070016	844038144	303575040	74649600	11943936	1119744	46656																	9474296896
7	268435456	2818572288	13388218368	38050725888	72137834496	96315899904	93244096512	66418900992	34966536192	13544423424	3804143616	752467968	99283968	7838208	279936															435817657216
8	4294967296	51539607552	283467841536	947040288768	2147752083456	3500666781696	4236314148864	3878456131584	2712982781952	1454421049344	595731677184	184605474816	42472636416	7023034368	788299776	53747712	1679616													20047612231936
9	68719476736	927712935936	5798205849600	22265110462464	58793807314944	113238960242688	164683541643264	184730838368256	161908556562432	111653060345856	60715708710912	25977774145536	8684483641344	2239344672768	436001439744	61917364224	6046617600	362797056	10077696											922190162669056
10	1099511627776	16492674416640	115448720916480	500964985405440	1509852803235840	3356465402216448	5705434556006400	7587532174786560	8012178275696640	6780639353241600	4622576127049728	2542739757465600	1126712570019840	400123767029760	112827197030400	24890780418048	4198771261440	522427760640	45148078080	2418647040	60466176									42420747482776576
11	17592186044416	290271069732864	2249600790429696	10885165114982400	36873496827002880	92945703625555968	180901239368712192	278469736554037248	344528231942062080	346307701317304320	284769632164773888	192309499288092672	106788612061790208	48699520497745920	18168480981319680	5506847817596928	1341522252398592	258474036953088	38453263073280	4256818790400	329903456256	15963070464	362797056							1951354384207722496
12	281474976710656	5066549580791808	43065671436730368	229894687228428288	864717516734201856	2437406172546859008	5346357697874755584	9354819751467024384	13281132824582160384	15480419207010582528	14934868337904058368	11989269368467881984	8032663568891510784	4495976013175455744	2100215860017758208	816350231619698688	262639589548621824	69373205554397184	14867762610438144	2541831636123648	338161684709376	33714004819968	2368339181568	104485552128	2176782336					89762301673555234816
13	4503599627370496	87820192733724672	812336782786953216	4742290407621132288	19611346789849890816	61131087285741158400	149227776407525916672	292614137807747678208	469091945415771684864	622562813610943315968	690174751322211876864	643147665247247007744	505874291972917690368	336655158075010842624	189702859489844133888	90442640425394110464	36395934151757266944	12311422827755470848	3478678672364273664	813734842154876928	155621121884946432	23906294326886400	2875999650840576	260795938111488	16752516857856	679156088832	13060694016			4129065876983540801536
14	72057594037927936	1513209474796486656	15132094747964866560	95899650465227341824	432470539117227147264	1477223461973982707712	3972458598117301813248	8630079362419031801856	15423577092483446734848	22974896534600099561472	28798855220312659722240	30589933557354697064448	27670581145433882492928	21386351528033680097280	14150215297475272507392	8019881823012630036480	3891175473576639725568	1613141027438626603008	569508611534503280640	170376582577753423872	42891646196061831168	8999816687398158336	1553493789568401408	216634492591276032	23783140689444864	1977702530678784	117023818383360	4388393189376	78364164096	189937030341242876870656

Technical details

Node id: n-524d4090108719a6 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-cb1c90404b4cec63

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           1;
  1:          16,           24,             6;
  2:         256,          768,           768,            288,             36;
  3:        4096,        18432,         32256,          27648,          12096,            2592,             216;
  4:       65536,       393216,        983040,        1327104,        1050624,          497664,          138240,           20736,            1296;
  5:     1048576,      7864320,      25559040,       47185920,       54558720,        41140224,        20459520,         6635520,         1347840,          155520,           7776;
  6:    16777216,    150994944,     603979776,     1415577600,     2158755840,      2250768384,      1642070016,       844038144,       303575040,        74649600,       11943936,        1119744,         46656;
  7:   268435456,   2818572288,   13388218368,    38050725888,    72137834496,     96315899904,     93244096512,     66418900992,     34966536192,     13544423424,     3804143616,      752467968,      99283968,       7838208,       279936;
  8:  4294967296,  51539607552,  283467841536,   947040288768,  2147752083456,   3500666781696,   4236314148864,   3878456131584,   2712982781952,   1454421049344,   595731677184,   184605474816,   42472636416,    7023034368,    788299776,    53747712,    1679616;
  9: 68719476736, 927712935936, 5798205849600, 22265110462464, 58793807314944, 113238960242688, 164683541643264, 184730838368256, 161908556562432, 111653060345856, 60715708710912, 25977774145536, 8684483641344, 2239344672768, 436001439744, 61917364224, 6046617600, 362797056, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (16+24*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(16 + 24*x + 6*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (16 + 24*x + 6*x^2)*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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 read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272867 → basis A013611

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.