A272867 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 36\,x + 76\,x^{2} + 68\,x^{3} + 22\,x^{4}\right)\,P_{n-1}(x)\)

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

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	6	24	22																											52
2	36	288	840	1056	484																									2704
3	216	2592	12744	32832	46728	34848	10648																							140608
4	1296	20736	143424	559872	1348704	2052864	1928256	1022208	234256																					7311616
5	7776	155520	1386720	7257600	24684480	57003264	90509760	97574400	68360160	28110720	5153632																			380204032
6	46656	1119744	12223872	80248320	352797120	1094114304	2454195456	4011752448	4743161280	3955944960	2209502592	742123008	113379904																	19770609664
7	279936	7838208	101243520	799497216	4311667584	16797279744	48745452672	107041720320	178733326464	225830094336	212549242752	144511589376	67100288640	19047823872	2494357888															1028071702528
8	1679616	53747712	801736704	7399268352	47285296128	221852639232	790464116736	2181681340416	4713801205248	7999498248192	10627350902784	10936513437696	8546963217408	4903948836864	1948320270336	478916714496	54875873536													53459728531456
9	10077696	362797056	6137316864	64819740672	478932424704	2628746846208	11107524556800	36941984907264	98033369969664	209125207904256	359455689888768	496664463753216	547559821670400	475154105868288	317417884489728	157520061186048	54686302336512	11853188683776	1207269217792											2779905883635712
10	60466176	2418647040	45752739840	544195584000	4564289295360	28692893564928	140271584993280	546064929423360	1719173397841920	4420275794657280	9332473268367360	16207677913743360	23113331237652480	26918978557870080	25354521924526080	19016539923972096	11091792741258240	4849031734272000	1494818795120640	289744612270080	26559922791424									144555105949057024
11	362797056	15963070464	333894223872	4416449495040	41452990064640	293672607326208	1630673320015872	7274151274807296	26506573563064320	79795244763463680	199903273434501120	418471493405736960	732978669259837440	1072802735153233920	1306675904164392960	1314825294005600256	1080747196962453504	713661094523215872	369365114700933120	144292816910499840	39999243723884544	7011819616935936	584318301411328							7516865509350965248
12	2176782336	104485552128	2394460569600	34863345893376	362058401193984	2854069294399488	17744148277346304	89241864690401280	369493885014110208	1274960562140086272	3697650133317550080	9064002907658649600	18839059872855736320	33234677328081715200	49712851792380395520	62850833637350178816	66787160129525772288	59146055762361384960	43120495289936756736	25431063733796143104	11829055591615832064	4176494739105841152	1051772942540390400	168283670806462464	12855002631049216					390877006486250192896
13	13060694016	679156088832	16922305880064	268945811177472	3061748841136128	26580811004706816	182927837555933184	1024100271275507712	4749139324755910656	18480937636367990784	60913819761577795584	171176753012654997504	411918731953928110080	850953564885518254080	1510368683831069736960	2301376346059028299776	3002825707505927626752	3340486517704490778624	3147545832885860769792	2488691496133209489408	1629970759539411075072	868439704776930164736	366785734409330835456	118135136906136649728	27254942851029983232	4010760820887355392	282810057883082752			20325604337285010030592
14	78364164096	4388393189376	118120916680704	2034751642140672	25197483244437504	238883646061412352	1802864963953557504	11119647571098992640	57097654885378179072	247395175580410970112	913506486986928586752	2895572820353892876288	7919858498782000103424	18757659522485794111488	38547309763225274941440	68778084915781245075456	106478097594735779168256	142741015699667830308864	165119116987353351094272	163963952355558712541184	138754428787922431131648	99081149372538055557120	58902624030969352617984	28617341952353223573504	11068059099858737872896	3277156205285050023936	697563862771254657024	95024179448715804672	6221821273427820544	1056931425538820521590784

Technical details

Node id: n-25060c57f9abbca7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-38e8cf17451719bf

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:        6,        24,         22;
  2:       36,       288,        840,        1056,          484;
  3:      216,      2592,      12744,       32832,        46728,         34848,          10648;
  4:     1296,     20736,     143424,      559872,      1348704,       2052864,        1928256,        1022208,         234256;
  5:     7776,    155520,    1386720,     7257600,     24684480,      57003264,       90509760,       97574400,       68360160,        28110720,         5153632;
  6:    46656,   1119744,   12223872,    80248320,    352797120,    1094114304,     2454195456,     4011752448,     4743161280,      3955944960,      2209502592,       742123008,       113379904;
  7:   279936,   7838208,  101243520,   799497216,   4311667584,   16797279744,    48745452672,   107041720320,   178733326464,    225830094336,    212549242752,    144511589376,     67100288640,     19047823872,      2494357888;
  8:  1679616,  53747712,  801736704,  7399268352,  47285296128,  221852639232,   790464116736,  2181681340416,  4713801205248,   7999498248192,  10627350902784,  10936513437696,   8546963217408,   4903948836864,   1948320270336,    478916714496,    54875873536;
  9: 10077696, 362797056, 6137316864, 64819740672, 478932424704, 2628746846208, 11107524556800, 36941984907264, 98033369969664, 209125207904256, 359455689888768, 496664463753216, 547559821670400, 475154105868288, 317417884489728, 157520061186048, 54686302336512, 11853188683776, 1207269217792;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+36*x+76*x^2+68*x^3+22*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[((6 + 36*x + 76*x^2 + 68*x^3 + 22*x^4)*P[n - 1, x])/(1 + 2*x + 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] = ((6 + 36*x + 76*x^2 + 68*x^3 + 22*x^4)*v[i - 1])/(1 + 2*x + 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 [-1, 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_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)

Related ?

Same fingerprint ?

A272867 → basis A013610 → 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.