Real-rooted atlas

A136124 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
26171235
3249813360315
4120634125310983603465
5720464411956153599846252045045
65040382681208442031611917949640820160675675
7403203515041311164271288433628012497410102895218144011486475
836288035667361531551637527596573940495610470034234668119224801814400218243025
93628800396590401923892565437466369867440461192381539959498380495806148149291280199584004583103525
1039916800479795040259219029682898840681737914413824957125481248630606481696784818875918778562011037760239500800105411381075
114790016006276458880373436190721331770826643163182225405254143795666227993590295268339666803116845607161228268652482901750048031135104002635284526875
126227020800882999878405733374722562254112741640597661605031611258842049918154521929813011556803259324611427574822828596033309824820968036197124467206419204358914560071152682225625
138717829120013296051417609351224429184401576404668001174843158290242472730294535923852133324869844497353510249593935065364182822548582857178921187602471496883770614328256073110576672006538371840002063427784543125
14130767436800021338929785600161542048705920751984197868944240478698490416055888444950682649734568458562232129094435851661291309836366267357410118978868460892585913627873159224657561036334087129641575289601267844804352001046139494400063966261320836875
Technical details
Node id
n-34e1cb08a06c788b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2d1227ef2afb39a7

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:       2,        3;
  2:       6,       17,        12;
  3:      24,       98,       133,        60;
  4:     120,      634,      1253,      1098,       360;
  5:     720,     4644,     11956,     15359,      9846,       2520;
  6:    5040,    38268,    120844,    203161,    191794,      96408,     20160;
  7:   40320,   351504,   1311164,   2712884,   3362801,    2497410,   1028952,    181440;
  8:  362880,  3566736,  15315516,  37527596,  57394049,   56104700,  34234668,  11922480,   1814400;
  9: 3628800, 39659040, 192389256, 543746636, 986744046, 1192381539, 959498380, 495806148, 149291280, 19958400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+3)*x+(n+2)*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 + 1 + (2*n + 3)*x + (n + 2)*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 + 1 + (2*n + 3)*x + (n + 2)*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

Candidate equalities ?

Same fingerprint ?

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.