Real-rooted atlas

A125092 → D+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-6 - 12\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 5\,x\)
\(P_{2}(x) = 9 + 26\,x + 9\,x^{2}\)
\(P_{3}(x) = 16 + 75\,x + 66\,x^{2} + 13\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
2926944
316756613170
42516424612417576
536305660570200211792
649510145518201095294255248
76479128144655406018694062914688
881116049561024811830789629405363339680
91001629813620244292322583013944435668437104192
101212210126453684064050708125061022920616585041267264
111442915188106286512804017047815246091410356408415103445672256
1216937562699410186023809537144840009230016815493553020111541236491662976
1319647453759615815841756074774794380085285255083624953576076144301456534055040
142255894510512369646976971411410204504321655921684683954954385385105924182911694579764864
Technical details
Node id
n-b4e86a61f3fb6cf1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-cb516ecd7bb869fb

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   4,    5;
  2:   9,   26,    9;
  3:  16,   75,   66,    13;
  4:  25,  164,  246,   124,    17;
  5:  36,  305,  660,   570,   200,    21;
  6:  49,  510, 1455,  1820,  1095,   294,    25;
  7:  64,  791, 2814,  4655,  4060,  1869,   406,   29;
  8:  81, 1160, 4956, 10248, 11830,  7896,  2940,  536,  33;
  9: 100, 1629, 8136, 20244, 29232, 25830, 13944, 4356, 684, 37;
%F Formula
The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 5*x, P(2,x) = 9 + 26*x + 9*x^2, P(3,x) = 16 + 75*x + 66*x^2 + 13*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 5*x;
P[2, x_] := 9 + 26*x + 9*x^2;
P[3, x_] := 16 + 75*x + 66*x^2 + 13*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 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; v[2] = 4 + 5*x; v[3] = 9 + 26*x + 9*x^2; v[4] = 16 + 75*x + 66*x^2 + 13*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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.