Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-4 - 12\,x - 12\,x^{2} - 4\,x^{3} - n^{2} - 3\,n^{2} x - 3\,n^{2} x^{2} - n^{2} x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n}(x) = \left(-16 - 60\,x - 66\,x^{2} + 2\,x^{3} + 42\,x^{4} + 18\,x^{5} - 4\,n - 24\,n x - 57\,n x^{2} - 67\,n x^{3} - 39\,n x^{4} - 9\,n x^{5} + 2\,n^{2} + 3\,n^{2} x - 12\,n^{2} x^{2} - 34\,n^{2} x^{3} - 30\,n^{2} x^{4} - 9\,n^{2} x^{5} + 2\,n^{3} + 9\,n^{3} x + 15\,n^{3} x^{2} + 11\,n^{3} x^{3} + 3\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 48\,x^{2} + 114\,x^{3} + 134\,x^{4} + 78\,x^{5} + 18\,x^{6} + 4\,n x + 24\,n x^{2} + 57\,n x^{3} + 67\,n x^{4} + 39\,n x^{5} + 9\,n x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 6\,x\)
\(P_{2}(x) = 14 + 42\,x + 31\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
214423187
344198294144680
4128768171616926214925
535226407880117008640253833750
69288352312006192068850406629963221875
7236824864111552277200412020366282180306379081412500

All 15 rows

Technical details
Node id
n-47dbbceb54d8e6b2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e6bf57180a08ced6

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:     4,      6;
  2:    14,     42,      31;
  3:    44,    198,     294,     144;
  4:   128,    768,    1716,    1692,     621;
  5:   352,   2640,    7880,   11700,    8640,     2538;
  6:   928,   8352,   31200,   61920,   68850,    40662,     9963;
  7:  2368,  24864,  111552,  277200,  412020,   366282,   180306,   37908;
  8:  5888,  70656,  370048, 1104768, 2056320,  2443392,  1809864,  763992,  140697;
  9: 14336, 193536, 1158912, 4040064, 9035712, 13444704, 13308624, 8450568, 3123036, 511758;
%F Formula
The row polynomials satisfy (n^3-n^2-4+(3*n^3-3*n^2-12)*x+(3*n^3-3*n^2-12)*x^2+(n^3-n^2-4)*x^3)*P(n,x) = (2*n^3+2*n^2-4*n-16+(9*n^3+3*n^2-24*n-60)*x+(15*n^3-12*n^2-57*n-66)*x^2+(11*n^3-34*n^2-67*n+2)*x^3+(3*n^3-30*n^2-39*n+42)*x^4-(9*n^2+9*n-18)*x^5)*P(n-1,x) + ((4*n+8)*x+(24*n+48)*x^2+(57*n+114)*x^3+(67*n+134)*x^4+(39*n+78)*x^5+(9*n+18)*x^6)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 6*x, P(2,x) = 14 + 42*x + 31*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[2, x_] := 14 + 42*x + 31*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 2*n^2 - 4*n - 16 + (9*n^3 + 3*n^2 - 24*n - 60)*x + (15*n^3 - 12*n^2 - 57*n - 66)*x^2 + (11*n^3 - 34*n^2 - 67*n + 2)*x^3 + (3*n^3 - 30*n^2 - 39*n + 42)*x^4 - (9*n^2 + 9*n - 18)*x^5)*P[n - 1, x] + ((4*n + 8)*x + (24*n + 48)*x^2 + (57*n + 114)*x^3 + (67*n + 134)*x^4 + (39*n + 78)*x^5 + (9*n + 18)*x^6)*D[P[n - 1, x], x])/(n^3 - n^2 - 4 + (3*n^3 - 3*n^2 - 12)*x + (3*n^3 - 3*n^2 - 12)*x^2 + (n^3 - n^2 - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; v[3] = 14 + 42*x + 31*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 + 2*n^2 - 4*n - 16 + (9*n^3 + 3*n^2 - 24*n - 60)*x + (15*n^3 - 12*n^2 - 57*n - 66)*x^2 + (11*n^3 - 34*n^2 - 67*n + 2)*x^3 + (3*n^3 - 30*n^2 - 39*n + 42)*x^4 - (9*n^2 + 9*n - 18)*x^5)*v[i - 1] + ((4*n + 8)*x + (24*n + 48)*x^2 + (57*n + 114)*x^3 + (67*n + 134)*x^4 + (39*n + 78)*x^5 + (9*n + 18)*x^6)*deriv(v[i - 1], x))/(n^3 - n^2 - 4 + (3*n^3 - 3*n^2 - 12)*x + (3*n^3 - 3*n^2 - 12)*x^2 + (n^3 - n^2 - 4)*x^3)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.