Real-rooted atlas

A269944 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} - 2\,n - 4\,n x - 2\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
22101224
3106413084288
41007602068240010086336
517001482049596800926273619152228096
644200432920170445634710803873712225456053625612317184
71635400177418407994875219490374427869946423449348810776931220913984936105984
8817700009721328004920013280139025223042406996788826217046528175821269766649703424108752716895482810368
9531505000068667222000384933760800123330483952024960169070883316795872816289938037081616102332299525162193953287286432025612603730968576
104358341000006077176404000373326150336001334654327478403082709929008964816426817663045163600437234563755870760046081775895817328644933730346854461206029015042092219340783616
114401924410000065868572910400043965432880056001732526805599264044882309856018016803978231271889921017615606379983689111937917998137656622016583493888232747945688079365699923150311936630422098854912426812745519857664
1253703477802000008581804521908800061805531154557920026588540705437964807623975041377285312153739408436632910082238424046560723590423734997779069525504182066890415039449609860654993747810304358146515087024025678370196669897932878172340258009088104995935397884985344
137787004281290000001323305768045716000010223327282131492000047638797102605063200014963279293696444528003349945753413638137024550568417153155298425667320580264093448759686129014584541292268288410617826319451240320019688287309571129518086401121623492773201921265391784421612789761149133401792733593630658813136182415720448
14132379072781930000000238355627931590520000019655742300662167920000985700779996947736800003363144790093202656800008268591819318973291756801512156978747509766891648209142754199301574220659221998464599144270042754561752240813284169378689536104096354553216533515776044745760932400057214361613161095225924311649894423718265475099387092992197650945108350178099210485314092574386176393216
Technical details
Node id
n-5dc372934f3fd998 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-41abbd63fc8edcc4

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,           3;
  2:          2,          10,           12;
  3:         10,          64,          130,            84;
  4:        100,         760,         2068,          2400,          1008;
  5:       1700,       14820,        49596,         80092,         62736,         19152;
  6:      44200,      432920,      1704456,       3471080,       3873712,       2254560,        536256;
  7:    1635400,    17741840,     79948752,     194903744,     278699464,     234493488,     107769312,      20913984;
  8:   81770000,   972132800,   4920013280,   13902522304,   24069967888,   26217046528,   17582126976,    6649703424,   1087527168;
  9: 5315050000, 68667222000, 384933760800, 1233304839520, 2496016907088, 3316795872816, 2899380370816, 1610233229952, 516219395328, 72864320256;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n^2-2*n+2+(2*n^2-4*n+6)*x+(n^2-2*n+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[Together[((n^2 - 2*n + 2 + (2*n^2 - 4*n + 6)*x + (n^2 - 2*n + 4)*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] = ((n^2 - 2*n + 2 + (2*n^2 - 4*n + 6)*x + (n^2 - 2*n + 4)*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.