Real-rooted atlas

A038220 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 12\,x + 7\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15712
2257049144
31255257353431728
4625350073506860240120736
531252187561250857506002516807248832
6156251312504593758575009003755042101176492985984
77812576562532156257503125105043758823675411771582354335831808
83906254375000214375006002500010504375011764900082354300329417205764801429981696
919531252460937513781250045018750094539375013235512501235314500741188700259416045403536075159780352
109765625136718750861328125321562500078782812501323551250015441431250123531450006485401125201768035028247524961917364224
11488281257519531255263671875221074218756190078125012132553125016985574375016985574375011889902062555486209625155361386951977326743743008370688
1224414062541015625003158203125014738281250046425585937510399331250001698557437500203826892500017834853093751109724192500466084160850118639604580138412872018916100448256
13122070312522216796875186621093750957988281250335295898437584494566406251577231906250022081246687500231853090218751803301812812510098490151750385578714885089968366806596889010407106993205379072
146103515625119628906250108862304687560962890625002347071289062565717996093750138007791796875220812466875000270495271921875252462253793750176723577655625899683668065003148892838227567822307284906782230728491283918464548864
Technical details
Node id
n-bb9d1d7a965ec78a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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:       5,        7;
  2:      25,       70,        49;
  3:     125,      525,       735,       343;
  4:     625,     3500,      7350,      6860,      2401;
  5:    3125,    21875,     61250,     85750,     60025,      16807;
  6:   15625,   131250,    459375,    857500,    900375,     504210,     117649;
  7:   78125,   765625,   3215625,   7503125,  10504375,    8823675,    4117715,    823543;
  8:  390625,  4375000,  21437500,  60025000, 105043750,  117649000,   82354300,  32941720,   5764801;
  9: 1953125, 24609375, 137812500, 450187500, 945393750, 1323551250, 1235314500, 741188700, 259416045, 40353607;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5+12*x+7*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[((5 + 12*x + 7*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] = ((5 + 12*x + 7*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. [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.