Real-rooted atlas

A013622 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(8 + 31\,x + 23\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182331
264368529961
35124416126961216729791
4409647104203136389344279841923521
5327684710402708480778688011193640643634328629151
6262144452198432501760124590080268647360308944464148035889887503681
72097152422051843640197121744261120501475072086504449928290009784340482544727512614111
8167772163858759683882876928223265423368023601152018454282649626528031308821790882860878310985281852891037441
913421772834728837123993816268826791850803211553985658883321770876928636672751411278447178298885638390940232180115266146326439622160671
101073741824308700774403993816268803061925806080154053142118405314833403084812733455028224020919247546368022553563760928014409221291704041426511213649819628286980801
1185899345922716566814723905064796160336811838668801936668072345607795088991191042241088084967424460223446020096066157120365388806340057368349760364553298680111295280975791392725408476896404831
126871947673623708219473923748862204313635926596124672023240016868147201069040775934771235857409359478784883629016358584321587770888769331202028818357871923201749855833664533769146973675973699221914624432020321787662783788549761
135497558138882054712354406435443788113510437363659969658882685513060319232013897530087152025653273865334082764815316236283548794883302563448640208896527492773046700032060661668900370503684756426311506323584227912094093011338450403636193646738324417546297445042591
1443980465111041770213720719363308086890594304380429992418344963007774627557539841729470410845585408745834114677158707224505978053678071808616478510412838993921181583811624608071681698526729210374103041775732489629027471361276307726920863495045645207253688434689611592836324538749809756943935220796320321
Technical details
Node id
n-a02c289146470276 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f64c6031b140aec2

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:         8,         23;
  2:        64,        368,         529;
  3:       512,       4416,       12696,        12167;
  4:      4096,      47104,      203136,       389344,        279841;
  5:     32768,     471040,     2708480,      7786880,      11193640,       6436343;
  6:    262144,    4521984,    32501760,    124590080,     268647360,     308944464,     148035889;
  7:   2097152,   42205184,   364019712,   1744261120,    5014750720,    8650444992,    8290009784,    3404825447;
  8:  16777216,  385875968,  3882876928,  22326542336,   80236011520,  184542826496,  265280313088,  217908828608,   78310985281;
  9: 134217728, 3472883712, 39938162688, 267918508032, 1155398565888, 3321770876928, 6366727514112, 7844717829888, 5638390940232, 1801152661463;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (8+31*x+23*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[((8 + 31*x + 23*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] = ((8 + 31*x + 23*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.