A272867 → basis A013609 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(22 + 20\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

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	22	20	4																											46
2	484	880	576	160	16																									2116
3	10648	29040	32208	18560	5856	960	64																							97336
4	234256	851840	1331968	1168640	628864	212480	44032	5120	256																					4477456
5	5153632	23425600	47277120	55756800	42535680	21926400	7733760	1843200	284160	25600	1024																			205962976
6	113379904	618435840	1529223168	2265894400	2240029440	1556121600	778813440	282931200	74050560	13619200	1671168	122880	4096																	9474296896
7	2494357888	15873186560	46465146112	82907883520	100715428352	88098841600	57216445440	28025241600	10402990080	2912358400	605351936	90603520	9232384	573440	16384															435817657216
8	54875873536	399097262080	1349674377216	2816769105920	4059757678592	4284114616320	3423600345088	2113279590400	1018236395520	384232652800	113176870912	25749749760	4436590592	559677440	48758784	2621440	65536													20047612231936
9	1207269217792	9877657236480	37894285034496	90558796922880	151048748556288	186712751554560	177240530632704	132100616355840	78361193889792	37270964633600	14247489798144	4366962524160	1065307471872	204043714560	30012604416	3271557120	248905728	11796480	262144											922190162669056
10	26559922791424	241453843558400	1036056492359680	2789689861939200	5285825546833920	7490890693017600	8237741699235840	7197875178700800	5074920715223040	2915587565158400	1372308843790336	530106830028800	167765974056960	43262961254400	9002381475840	1488401203200	190957486080	18323865600	1237319680	52428800	1048576									42420747482776576
11	584318301411328	5843183014113280	27728559394246656	83060122184089600	176226185238568960	281674865630822400	352191433430876160	353071650688204800	288556726105866240	194432841452748800	108802228727447552	50770877397073920	19782223404990464	6427531948851200	1734375513784320	385844301004800	69978614661120	10175879577600	1157528289280	99195289600	6020923392	230686720	4194304							1951354384207722496
12	12855002631049216	140236392338718720	729229240161337344	2405266607991357440	5649092756507295744	10053611237385830400	14086613589049999360	15938104446281318400	14818446721816657920	11460943636830617600	7436532765482287104	4070735243095572480	1885835377761058816	740133680562831360	245835793900240896	68886212693196800	16193917597777920	3166818847948800	508897672560640	66036380467200	6746479263744	522274734080	28789702656	1006632960	16777216					89762301673555234816
13	282810057883082752	3342300684072796160	18899191140847992832	68061395748391485440	175302289763633004544	343782368784599613440	533574094732845776896	672585014548732313600	701114371161792839680	612262112231732019200	452096380464489299968	284130605205070807040	152649214234583891968	70282589499985756160	27754402588106162176	9392747279506472960	2717333616616013824	669093217383219200	139307796146421760	24298029213286400	3504740843388928	410565151293440	38064785195008	2687038914560	135660568576	4362076160	67108864			4129065876983540801536
14	6221821273427820544	79186816207263170560	483759459011644096512	1888703732017863720960	5293475054331147780096	11341503491527417528320	19315486618869131378688	26843481689867424890880	31010512835465471852544	30182393950528890470400	24995820099500576342016	17741849372728271831040	10849280339120218963968	5735723674511647703040	2626845503876386258944	1042858849911208673280	358653895508106412032	106637712232776990720	27315970329349718016	5997085435153612800	1120296067405971456	176319367049379840	23067691673714688	2462671165194240	208984451579904	13557332705280	631360192512	18790481920	268435456	189937030341242876870656

Technical details

Node id: n-c6c043a068fbd531 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9217fe7884f21aef

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:            22,            20,              4;
  2:           484,           880,            576,            160,              16;
  3:         10648,         29040,          32208,          18560,            5856,             960,              64;
  4:        234256,        851840,        1331968,        1168640,          628864,          212480,           44032,            5120,            256;
  5:       5153632,      23425600,       47277120,       55756800,        42535680,        21926400,         7733760,         1843200,         284160,          25600,           1024;
  6:     113379904,     618435840,     1529223168,     2265894400,      2240029440,      1556121600,       778813440,       282931200,       74050560,       13619200,        1671168,        122880,          4096;
  7:    2494357888,   15873186560,    46465146112,    82907883520,    100715428352,     88098841600,     57216445440,     28025241600,    10402990080,     2912358400,      605351936,      90603520,       9232384,       573440,       16384;
  8:   54875873536,  399097262080,  1349674377216,  2816769105920,   4059757678592,   4284114616320,   3423600345088,   2113279590400,  1018236395520,   384232652800,   113176870912,   25749749760,    4436590592,    559677440,    48758784,    2621440,     65536;
  9: 1207269217792, 9877657236480, 37894285034496, 90558796922880, 151048748556288, 186712751554560, 177240530632704, 132100616355840, 78361193889792, 37270964633600, 14247489798144, 4366962524160, 1065307471872, 204043714560, 30012604416, 3271557120, 248905728, 11796480, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (22+20*x+4*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[(22 + 20*x + 4*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] = (22 + 20*x + 4*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 (−∞, -1]
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), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 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)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 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) (above)

Related ?

Same fingerprint ?

A272867 → basis A013610 → x ↦ x/(x+1)

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.