Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(-4 - 4\,x + n + 2\,n x\right)\,P_{n}(x) = \left(6 + 40\,x + 111\,x^{2} + 130\,x^{3} + 66\,x^{4} + 12\,x^{5} - 2\,n - 22\,n x - 90\,n x^{2} - 146\,n x^{3} - 97\,n x^{4} - 22\,n x^{5} + 2\,n^{2} x + 18\,n^{2} x^{2} + 44\,n^{2} x^{3} + 40\,n^{2} x^{4} + 12\,n^{2} x^{5} - n^{3} x^{2} - 4\,n^{3} x^{3} - 5\,n^{3} x^{4} - 2\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(3\,x + 20\,x^{2} + 45\,x^{3} + 46\,x^{4} + 22\,x^{5} + 4\,x^{6} - n x - 11\,n x^{2} - 34\,n x^{3} - 45\,n x^{4} - 27\,n x^{5} - 6\,n x^{6} + n^{2} x^{2} + 5\,n^{2} x^{3} + 9\,n^{2} x^{4} + 7\,n^{2} x^{5} + 2\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-11 - 50\,x - 58\,x^{2} - 20\,x^{3} + 3\,n + 24\,n x + 41\,n x^{2} + 18\,n x^{3} - 2\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 12\,x^{3} - 4\,x^{4} + n x + 4\,n x^{2} + 5\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
251015
3104645101
419152327208706
5364481593217610155268
669126266121456414380526242149
7134349825413803561260409667828980361099
826396769399340068488815510713846684651691763301796
95202690834142118873085547365935872891171574777240104377532100422
101033755061232118860460032229115713268009646459278497504353631876785700330580155
112058214102444754538529256178999980499196076878971331987539424688031694271323240463495673593604273
124107613472161226751710022649665377103304615320727535708210582574296101367659166161599740215735075633173324441104276582
1382041774680588339217567542805129880475210802800725644155617810206502133612602988298810496931158056497664520177663754202481691186493279034840
14163975176342216406200335250230026960888950131189921154418671448188915349818248139900828437914969884672701101022064274531045333012151413529449193614196306194585275793
Technical details
Node id
n-801c96cbd9fb40ce · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f6a909163e19896b

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:   5,    10;
  3:  10,    46,     45;
  4:  19,   152,    327,     208;
  5:  36,   448,   1593,    2176,    1015;
  6:  69,  1262,   6612,   14564,   14380,    5262;
  7: 134,  3498,  25413,   80356,  126040,   96678,   28980;
  8: 263,  9676,  93993,  400684,  888155, 1071384,  668465,  169176;
  9: 520, 26908, 341421, 1887308, 5547365, 9358728, 9117157, 4777240, 1043775;
%F Formula
The row polynomials satisfy (n-4+(2*n-4)*x)*P(n,x) = -(2*n-6-(2*n^2-22*n+40)*x+(n^3-18*n^2+90*n-111)*x^2+(4*n^3-44*n^2+146*n-130)*x^3+(5*n^3-40*n^2+97*n-66)*x^4+(2*n^3-12*n^2+22*n-12)*x^5)*P(n-2,x) - ((n-3)*x-(n^2-11*n+20)*x^2-(5*n^2-34*n+45)*x^3-(9*n^2-45*n+46)*x^4-(7*n^2-27*n+22)*x^5-(2*n^2-6*n+4)*x^6)*d/dx P(n-2,x) + (3*n-11-(2*n^2-24*n+50)*x-(6*n^2-41*n+58)*x^2-(4*n^2-18*n+20)*x^3)*P(n-1,x) + ((n-4)*x+(4*n-12)*x^2+(5*n-12)*x^3+(2*n-4)*x^4)*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[(-(2*n - 6 - (2*n^2 - 22*n + 40)*x + (n^3 - 18*n^2 + 90*n - 111)*x^2 + (4*n^3 - 44*n^2 + 146*n - 130)*x^3 + (5*n^3 - 40*n^2 + 97*n - 66)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*P[n - 2, x] - ((n - 3)*x - (n^2 - 11*n + 20)*x^2 - (5*n^2 - 34*n + 45)*x^3 - (9*n^2 - 45*n + 46)*x^4 - (7*n^2 - 27*n + 22)*x^5 - (2*n^2 - 6*n + 4)*x^6)*D[P[n - 2, x], x] + (3*n - 11 - (2*n^2 - 24*n + 50)*x - (6*n^2 - 41*n + 58)*x^2 - (4*n^2 - 18*n + 20)*x^3)*P[n - 1, x] + ((n - 4)*x + (4*n - 12)*x^2 + (5*n - 12)*x^3 + (2*n - 4)*x^4)*D[P[n - 1, x], x])/(n - 4 + (2*n - 4)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 6 - (2*n^2 - 22*n + 40)*x + (n^3 - 18*n^2 + 90*n - 111)*x^2 + (4*n^3 - 44*n^2 + 146*n - 130)*x^3 + (5*n^3 - 40*n^2 + 97*n - 66)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*v[i - 2] - ((n - 3)*x - (n^2 - 11*n + 20)*x^2 - (5*n^2 - 34*n + 45)*x^3 - (9*n^2 - 45*n + 46)*x^4 - (7*n^2 - 27*n + 22)*x^5 - (2*n^2 - 6*n + 4)*x^6)*deriv(v[i - 2], x) + (3*n - 11 - (2*n^2 - 24*n + 50)*x - (6*n^2 - 41*n + 58)*x^2 - (4*n^2 - 18*n + 20)*x^3)*v[i - 1] + ((n - 4)*x + (4*n - 12)*x^2 + (5*n - 12)*x^3 + (2*n - 4)*x^4)*deriv(v[i - 1], x))/(n - 4 + (2*n - 4)*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

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.