Real-rooted atlas

A008277 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2325
3712322
4155030499
531180195605471
663602105056010562386
71271932510342001330168712867
825560502331027804132302772252873681
9511186601023151701001141353528052503609446620
10102357002437250986920897435383922831609240495102856457
11204717305218345035517600661826037643761113189178200153456601119217243
12409552325075975903003400446606560342925441329728428760163539252431085812135610448
138191157794031175235160300140317181865295975680146386240410810406774768660660370371092131001159901
14163834750202127067850843067680210346636524524479981516394880537027920113963850147947801171170546001365147714225057

Other recurrences

Image under 1+D · proved equal for all n (proof step)

\(P_{n}(x) = \left(2\,x + 2\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(x^{2} + x^{3}\right)\,P_{n-3}'(x) + \left(-2 - 5\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(-x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)
Technical details
Node id
n-8d7a43e42ddb80a0 · revision 0 · JSON
Definition source
derived:derivative
derived:one_plus_derivative (equal by e-13e5451e4bd6e7bd)
Fingerprint
f6-90c2a235bbf6b62e

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:   1;
  2:   3,     2;
  3:   7,    12,      3;
  4:  15,    50,     30,      4;
  5:  31,   180,    195,     60,      5;
  6:  63,   602,   1050,    560,    105,     6;
  7: 127,  1932,   5103,   4200,   1330,   168,    7;
  8: 255,  6050,  23310,  27804,  13230,  2772,  252,   8;
  9: 511, 18660, 102315, 170100, 114135, 35280, 5250, 360, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + x*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 2*x + x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + x*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.