Real-rooted atlas

A038208 → D → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
216824
39612024240
45121216576642368
52560104008800224016024160
61228881024109440480007680384258816
7573445958401210496815360219520241928962923648
826214442168321242931212085248487424088883271680204834830336
9117964829053440121351680163924992929940482496614432901122027524608436967424
10524288019631104011432448002091059200160419840059501568011397120011366400552960102405760972800
112306867213071872001049243852825516789760257757491201269247795233149276164757913603717120014643202252879637088256
12100663296860646604894422417408301257572352393100984320250086752256855532339201657538150418490982401162444803784704491521151672647680
13436207616561601331208369749606403468205604864576303753625646447012085762024777351168508585279488759771217926776872960350265344958464010649617385992232960
14187904819236382038425673318214860803915262181376081946158694400824404990689284489244311552014219788943360273184476364832420279091223649353728102278758423855104229376273429776334848
Technical details
Node id
n-28dad9d39eaca9ff · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       2;
  2:      16,        8;
  3:      96,      120,        24;
  4:     512,     1216,       576,        64;
  5:    2560,    10400,      8800,      2240,      160;
  6:   12288,    81024,    109440,     48000,     7680,      384;
  7:   57344,   595840,   1210496,    815360,   219520,    24192,     896;
  8:  262144,  4216832,  12429312,  12085248,  4874240,   888832,   71680,   2048;
  9: 1179648, 29053440, 121351680, 163924992, 92994048, 24966144, 3290112, 202752, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+2*n*x)*P(n-1,x) + 2*n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 2*n*x)*P[n - 1, x] + 2*n*x*D[P[n - 1, x], x])/(n - 1)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 2*n*x)*v[i - 1] + 2*n*x*deriv(v[i - 1], x))/(n - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.