Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-2 - x + x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
24232855
328201426280935
4280237468738338364021505
536403534212733321836218072858240623645
6582406346322708826591311970405264365672110656021827575
71106560133392886542959817194343326385861223783934011706942424344320894930575
8243443203211283361772933356541849547610103475289118289307808521510828346231064060860800042061737025
9608608000870984936053314927308185104520868404304755553578620577592544247832540325160069184112159897920170410240002229272062325
10170410240002627426300801762823294784683568933085217058773302392287348235947193317617721647225976164745892132204392864643954105507520528271744000131527051677175
115282717440008724416348480635807715610242718423612790687612354096231201470777823717617200544549593107819332511324827001293023219240712572072206472896150816011319680179612392960008549258359016375
1217961239296000316176210376320248454963796857611595128831246200359401712945115967817215616245441812260392633920848113993202185386168611549308135604410867292314131984808262944160243662726190874554892160664565853952000606997343490162625
13664565853952000124169693557638401045753850198901125284017522748524401793591491146777052432997662979127730676632315210488905171008164186216122162298705248843280994367109538877125202216366458595818094438412808389995056608002722239187919104002658263415808000046738795448742522125
142658263415808000052525510595048960047169450830934496002563281164684937241694464934993689737000250323499310962505476492718255922980544838732784629891551157111828331595446056507186708814125111116364388452293610043621470648208810756182007040512659650337304198304001276893387437534720011430532687974400003879320022245629336375
Technical details
Node id
n-290beb665c00091f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dcd638913f419d0f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         1,          4;
  2:         4,         23,          28;
  3:        28,        201,         426,          280;
  4:       280,       2374,        6873,         8338,         3640;
  5:      3640,      35342,      127333,       218362,       180728,        58240;
  6:     58240,     634632,     2708826,      5913119,      7040526,      4365672,      1106560;
  7:   1106560,   13339288,    65429598,    171943433,    263858612,    237839340,    117069424,     24344320;
  8:  24344320,  321128336,  1772933356,   5418495476,  10103475289,  11828930780,   8521510828,   3462310640,    608608000;
  9: 608608000, 8709849360, 53314927308, 185104520868, 404304755553, 578620577592, 544247832540, 325160069184, 112159897920, 17041024000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n-2+(6*n-1)*x+(3*n+1)*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[Together[((3*n - 2 + (6*n - 1)*x + (3*n + 1)*x^2)*P[n - 1, x])/(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] = ((3*n - 2 + (6*n - 1)*x + (3*n + 1)*x^2)*v[i - 1])/(1 + x)); 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

Proof chains ? JSON