Real-rooted atlas

A049020 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,x - x^{2} + 2\,n + 3\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(-2 - 5\,x - 4\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
267215
3224125593
4942432329715681
5454150919811284411525691
62430990316636147447271189220353079
71421468753141205159599107206427979402877544053
8899185041711226688169133914452217836352633065012741406058545
9610182389542110972405178951421862074312818567583760916958122851512114772652179
1044127983162679110133494019112123423494343719667562511353479344625108114291611722345115975931542207
113382797426911242596734321320736458812945113590290955580220400626411015128696351287388805127281099929767857012697268205
1227364652623939143399546410160229439519973701119463942210904174348973987682107094735892212580692852336332591926446739940044213597183092096409
132326980998222114819419736178207725947286722446902102761960717277435057909206998641196385787621858325188885428107907239032658164534277214522632566276444372781622021899
14207325040622144961962551025538976892300309386271660166603625718723646818314943896658533577417762792364836505819432229010601162180900274440820674376526636129691118386464973119089932244367455166279
Technical details
Node id
n-d1b016a8baf4eef6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c8d1d9d3f365f998

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:      2,       1;
  2:      6,       7,        2;
  3:     22,      41,       25,        5;
  4:     94,     243,      232,       97,       15;
  5:    454,    1509,     1981,     1284,      411,       52;
  6:   2430,    9903,    16636,    14744,     7271,     1892,     203;
  7:  14214,   68753,   141205,   159599,   107206,    42797,    9402,     877;
  8:  89918,  504171,  1226688,  1691339,  1445221,   783635,  263306,   50127,   4140;
  9: 610182, 3895421, 10972405, 17895142, 18620743, 12818567, 5837609, 1695812, 285151, 21147;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+(3*n-2)*x+(n-1)*x^2)*P(n-1,x) - (2+5*x+4*x^2+x^3)*d/dx 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[(2*n + (3*n - 2)*x + (n - 1)*x^2)*P[n - 1, x] - (2 + 5*x + 4*x^2 + x^3)*D[P[n - 1, x], 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] = (2*n + (3*n - 2)*x + (n - 1)*x^2)*v[i - 1] - (2 + 5*x + 4*x^2 + x^3)*deriv(v[i - 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. [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.