Real-rooted atlas

A038279 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(9\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
236844
348621624726
4583238888646410648
56561058320194402880160146410
67085887873203499207776086403841932612
774401749920232551124016329602721602419289624801854
87652750411904278479361856293932806531840870912645122048311794736
97748409781377495072107138505647617113613226976023514624261273616588846083858459858
1077484097801549681956013774950720714256704023808556805290790407838208074649604147201024047158953820
11767092568221704650151601704650151601010163052803928411872010475764992193995648024634368020528640101376022528570623341222
12753145430616184102216372820455801819201363720121280606097831680188563769856419030599686651279360739031040547430402433024491526847480094664
13734316794850619581781196016239332881284641772836157664088641807883203151708724736817109669376155639937024216166579202134978560142331904575078410649681599137794746
14711722431932122056087025581682741449367442242233773558656641240985310364804963941241459214707974048768326843867750454473977958467251824640597793996836229939213418496229376966635940030068
Technical details
Node id
n-da07574fa3209102 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0205823d5b324a40

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:         2;
  2:        36,          8;
  3:       486,        216,         24;
  4:      5832,       3888,        864,        64;
  5:     65610,      58320,      19440,      2880,       160;
  6:    708588,     787320,     349920,     77760,      8640,      384;
  7:   7440174,    9920232,    5511240,   1632960,    272160,    24192,     896;
  8:  76527504,  119042784,   79361856,  29393280,   6531840,   870912,   64512,   2048;
  9: 774840978, 1377495072, 1071385056, 476171136, 132269760, 23514624, 2612736, 165888, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (9*n+2*n*x)*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[((9*n + 2*n*x)*P[n - 1, 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] = ((9*n + 2*n*x)*v[i - 1])/(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

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.