A200536 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - x + 4\,x^{2} + 7\,n + 8\,n x\right)\,P_{n}(x) = \left(18 + 84\,x + 146\,x^{2} + 112\,x^{3} + 32\,x^{4} + 42\,n + 146\,n x + 168\,n x^{2} + 64\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	20	30	8																											58
2	204	752	964	480	64																									2464
3	1728	10296	24720	30200	19296	5760	512																							92512
4	13392	110592	391968	774784	926736	677632	288256	61440	4096																					3248896
5	98496	1041120	4890240	13408320	23682880	28013024	22293120	11627520	3706880	614400	32768																			109408768
6	699840	9020160	52809408	185448960	434260800	712608000	837344320	706263552	421002240	170639360	43499520	5898240	262144																	3579756544
7	4852224	73809792	517757184	2218043520	6476004864	13613314176	21208748288	24815796096	21837831680	14309349376	6817800192	2259582976	479854592	55050240	2097152															114689892352
8	33032448	579280896	4736517120	23955886080	83826925056	215009206272	417706791936	626123509760	730506461440	663951826944	466921570304	250110672896	99241820160	27818721280	5062524928	503316480	16777216													3616104841216
9	221709312	4403953152	41130436608	239889475584	978675664896	2964140282880	6902446731264	12625313605632	18368827610112	21387462516224	19939656812544	14813410590720	8674147958784	3928985174016	1335616339968	324362305536	51640270848	4529848320	134217728											112584995504128
10	1471343616	32651735040	342943994880	2266093117440	10561806120960	36894406238208	100178390476800	216390232719360	377389509073920	536199414231040	623380202898432	593244624609280	460489761177600	289248769474560	145083169505280	56924426993664	16925400760320	3625891921920	513047265280	40265318400	1073741824									3469733552717824
11	9674588160	237228963840	2767819051008	20436425349120	107134866432000	424051576565760	1315568423854080	3277760647163904	6666007281500160	11185186640517120	15587932329840640	18101850168227840	17521111203864576	14097115597537280	9372454848430080	5099010502164480	2236729440337920	774027211702272	204506545520640	39190905487360	4990215127040	354334801920	8589934592							106038442476961792
12	63126687744	1694987845632	21749925371904	177473870069760	1033646946545664	4572187575386112	15956140126519296	45049809728765952	104686506248417280	202622848065994752	329292775732174848	451608159047614464	523980711482626048	514429509645434880	426463547638284288	297169456328605696	172772352184025088	82894435569893376	32308280457953280	10000054496526336	2377294194671616	412114996953088	47699906789376	3092376453120	68719476736					3217881673379086336
13	409235079168	11941827895296	167147859640320	1493661278453760	9568894251810816	46780822689030144	181393875956367360	572380561328209920	1496300774185451520	3282004628241850368	6094771344838066176	9641985149153230848	13045828057432801280	15126012702260060160	15030949852202565632	12780143096614879232	9264411914768744448	5692878512728309760	2940561929829089280	1261795491063005184	442449132804636672	123868425016573952	26764163866951680	4235610847969280	449356658376704	26800595927040	549755813888			97067234807290789888
14	2638260191232	83135670976512	1260504993742848	12243021043728384	85562016731283456	458143219474956288	1954172247067901952	6815689718527033344	19797968424932425728	48535470783271600128	101398209542471761920	181791576590897774592	281067441169042784256	375919852063239766016	435618824446774362112	437412521458527240192	380073542370007449600	284965605928433876992	183519416774220054528	100850515772151169024	46863260473104531456	18187188499101253632	5795807414234120192	1481138842328104960	293305051810627584	42698709430960128	4181167842525184	230897441832960	4398046511104	2912945917279080546304

Technical details

Node id: n-5fedd6b839deb652 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a1bc1a919c7f051b

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:        20,         30,           8;
  2:       204,        752,         964,          480,           64;
  3:      1728,      10296,       24720,        30200,        19296,          5760,           512;
  4:     13392,     110592,      391968,       774784,       926736,        677632,        288256,          61440,           4096;
  5:     98496,    1041120,     4890240,     13408320,     23682880,      28013024,      22293120,       11627520,        3706880,         614400,          32768;
  6:    699840,    9020160,    52809408,    185448960,    434260800,     712608000,     837344320,      706263552,      421002240,      170639360,       43499520,        5898240,        262144;
  7:   4852224,   73809792,   517757184,   2218043520,   6476004864,   13613314176,   21208748288,    24815796096,    21837831680,    14309349376,     6817800192,     2259582976,     479854592,      55050240,       2097152;
  8:  33032448,  579280896,  4736517120,  23955886080,  83826925056,  215009206272,  417706791936,   626123509760,   730506461440,   663951826944,   466921570304,   250110672896,   99241820160,   27818721280,    5062524928,    503316480,    16777216;
  9: 221709312, 4403953152, 41130436608, 239889475584, 978675664896, 2964140282880, 6902446731264, 12625313605632, 18368827610112, 21387462516224, 19939656812544, 14813410590720, 8674147958784, 3928985174016, 1335616339968, 324362305536, 51640270848, 4529848320, 134217728;

%F Formula

The row polynomials satisfy (7*n-4+(8*n-1)*x+4*x^2)*P(n,x) = (42*n+18+(146*n+84)*x+(168*n+146)*x^2+(64*n+112)*x^3+32*x^4)*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[Together[((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*P[n - 1, x])/(7*n - 4 + (8*n - 1)*x + 4*x^2)]];
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] = ((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*v[i - 1])/(7*n - 4 + (8*n - 1)*x + 4*x^2)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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

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.