Real-rooted atlas

A124932 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-16 - 32\,x - 24\,x^{2} - 8\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(32 + 48\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-24 - 24\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 21 + 12\,x\)
\(P_{3}(x) = 96 + 108\,x + 30\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2211233
39610830234
4360600330601350
51200264021607801056885
6369610080109205880157516832319
710752349444704033600134402856252142884
82995211289618144016128085680272164788360603612
9806403456006451206854404536001915205040075604952460375
102112001013760215424026611202106720110880038808087120113856609743085
115406722872320684288096307208870400558835224393607286401425601650085837673262
1213578247907328208665603294720034594560253693441326124849420801287000223080231661092142780482
1333546242124595261501440107627520126846720106089984645765122882880093693602162160336336316681365531972441
14817152055910400176117760338257920442041600415134720288288000149909760583783201681680034834804914004231516801953045675
Technical details
Node id
n-488548c9031501e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5969d25088d3741c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:     0;
  1:     3;
  2:    21,     12;
  3:    96,    108,     30;
  4:   360,    600,    330,     60;
  5:  1200,   2640,   2160,    780,    105;
  6:  3696,  10080,  10920,   5880,   1575,    168;
  7: 10752,  34944,  47040,  33600,  13440,   2856,   252;
  8: 29952, 112896, 181440, 161280,  85680,  27216,  4788,  360;
  9: 80640, 345600, 645120, 685440, 453600, 191520, 50400, 7560, 495;
%F Formula
The row polynomials satisfy P(n,x) = -(16+32*x+24*x^2+8*x^3+x^4)*P(n-4,x) + (32+48*x+24*x^2+4*x^3)*P(n-3,x) - (24+24*x+6*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 21 + 12*x, P(3,x) = 96 + 108*x + 30*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 21 + 12*x;
P[3, x_] := 96 + 108*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[-(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*P[n - 4, x] + (32 + 48*x + 24*x^2 + 4*x^3)*P[n - 3, x] - (24 + 24*x + 6*x^2)*P[n - 2, x] + (8 + 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] = 0; v[2] = 3; v[3] = 21 + 12*x; v[4] = 96 + 108*x + 30*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*v[i - 4] + (32 + 48*x + 24*x^2 + 4*x^3)*v[i - 3] - (24 + 24*x + 6*x^2)*v[i - 2] + (8 + 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.