Real-rooted atlas

A027467 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 36\,x + 20\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1162036
22566404001296
340961536019200800046656
4655363276806144005120001600001679616
510485766553600163840002048000012800000320000060466176
616777216125829120393216000655360000614400000307200000640000002176782336
7268435456234881024088080384001835008000022937600000172032000007168000000128000000078364164096
8429496729642949672960187904819200469762048000734003200000734003200000458752000000163840000000256000000002821109907456
968719476736773094113280386547056640011274289152000211392921600002642411520000022020096000000117964800000003686400000000512000000000101559956668416
101099511627776137438953472007730941132800025769803776000056371445760000084557168640000088080384000000062914560000000029491200000000081920000000000102400000000003656158440062976
111759218604441624189255811072015118284881920005669356830720000141733920768000002480343613440000031004295168000000276824064000000001730150400000000072089600000000001802240000000000204800000000000131621703842267136
1228147497671065642221246506598402902710697328640012094627905536000034016140984320000068032281968640000099213744537600000010630044057600000008304721920000000004613734400000000001730150400000000003932160000000000040960000000000004738381338321616896
1345035996273704967318349394477056054887620458577920025156826043514880007861508138598400000176883933118464000002948065551974400000036850819399680000000345476431872000000002399141888000000000011995709440000000000408944640000000000085196800000000000081920000000000000170581728179578208256
147205759403792793612610078956637388801024568915226787840051228445761339392000176097782304604160000440244455761510400000825458354552832000000117922622078976000000012897786789888000000001074815565824000000000671759728640000000000305345331200000000000954204160000000000001835008000000000000016384000000000000006140942214464815497216
Technical details
Node id
n-82073868f14b935d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6b726760bd05165a

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:          16,           20;
  2:         256,          640,           400;
  3:        4096,        15360,         19200,           8000;
  4:       65536,       327680,        614400,         512000,         160000;
  5:     1048576,      6553600,      16384000,       20480000,       12800000,        3200000;
  6:    16777216,    125829120,     393216000,      655360000,      614400000,      307200000,       64000000;
  7:   268435456,   2348810240,    8808038400,    18350080000,    22937600000,    17203200000,     7168000000,     1280000000;
  8:  4294967296,  42949672960,  187904819200,   469762048000,   734003200000,   734003200000,   458752000000,   163840000000,   25600000000;
  9: 68719476736, 773094113280, 3865470566400, 11274289152000, 21139292160000, 26424115200000, 22020096000000, 11796480000000, 3686400000000, 512000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+36*x+20*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[((16 + 36*x + 20*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] = ((16 + 36*x + 20*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

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.