Real-rooted atlas

A352880 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(9 - 9\,x + 3\,x^{2} + 18\,n - 15\,n x + 7\,n x^{2} + 11\,n^{2} - n^{2} x + 5\,n^{2} x^{2} + 5\,n^{3} + 7\,n^{3} x + n^{3} x^{2} + 4\,n^{4} + 2\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(36 - 36\,x + 12\,x^{2} + 60\,n - 12\,n x - 4\,n x^{2} + 10\,n x^{3} + 49\,n^{2} + 23\,n^{2} x + 5\,n^{2} x^{2} + 11\,n^{2} x^{3} + 26\,n^{3} + 22\,n^{3} x + 17\,n^{3} x^{2} + 3\,n^{3} x^{3} + 8\,n^{4} + 12\,n^{4} x + 6\,n^{4} x^{2} + n^{5} + 3\,n^{5} x\right)\,P_{n-1}(x) + \left(12\,x - 18\,x^{2} + 12\,x^{3} - 6\,x^{4} + 18\,n x - 13\,n x^{2} + 2\,n x^{3} - 7\,n x^{4} + 14\,n^{2} x - 2\,n^{2} x^{2} - 10\,n^{2} x^{3} - 2\,n^{2} x^{4} + 8\,n^{3} x - 4\,n^{3} x^{2} - 4\,n^{3} x^{3} + 2\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
088
1301242
2969616208
325050021020980
455219201584384244464
510785880833039206302819866
6192015232336002688083209603286944
73186347761111321375927182015876138636375804

All 15 rows

Technical details
Node id
n-e214a00167564f8e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9f01c67d4f0b38b0

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:    8;
  1:   30,     12;
  2:   96,     96,     16;
  3:  250,    500,    210,      20;
  4:  552,   1920,   1584,     384,      24;
  5: 1078,   5880,   8330,    3920,     630,      28;
  6: 1920,  15232,  33600,   26880,    8320,     960,     32;
  7: 3186,  34776, 111132,  137592,   71820,   15876,   1386,    36;
  8: 5000,  72000, 315840,  564480,  453600,  168000,  28000,  1920,   40;
  9: 7502, 137940, 797148, 1951488, 2256408, 1280664, 355740, 46464, 2574, 44;
%F Formula
The row polynomials satisfy (n^5+4*n^4+5*n^3+11*n^2+18*n+9+(2*n^4+7*n^3-n^2-15*n-9)*x+(n^3+5*n^2+7*n+3)*x^2)*P(n,x) = (n^5+8*n^4+26*n^3+49*n^2+60*n+36+(3*n^5+12*n^4+22*n^3+23*n^2-12*n-36)*x+(6*n^4+17*n^3+5*n^2-4*n+12)*x^2+(3*n^3+11*n^2+10*n)*x^3)*P(n-1,x) + ((2*n^4+8*n^3+14*n^2+18*n+12)*x-(2*n^4+4*n^3+2*n^2+13*n+18)*x^2-(4*n^3+10*n^2-2*n-12)*x^3-(2*n^2+7*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8.
%t Mathematica
P[0, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((n^5 + 8*n^4 + 26*n^3 + 49*n^2 + 60*n + 36 + (3*n^5 + 12*n^4 + 22*n^3 + 23*n^2 - 12*n - 36)*x + (6*n^4 + 17*n^3 + 5*n^2 - 4*n + 12)*x^2 + (3*n^3 + 11*n^2 + 10*n)*x^3)*P[n - 1, x] + ((2*n^4 + 8*n^3 + 14*n^2 + 18*n + 12)*x - (2*n^4 + 4*n^3 + 2*n^2 + 13*n + 18)*x^2 - (4*n^3 + 10*n^2 - 2*n - 12)*x^3 - (2*n^2 + 7*n + 6)*x^4)*D[P[n - 1, x], x])/(n^5 + 4*n^4 + 5*n^3 + 11*n^2 + 18*n + 9 + (2*n^4 + 7*n^3 - n^2 - 15*n - 9)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = ((n^5 + 8*n^4 + 26*n^3 + 49*n^2 + 60*n + 36 + (3*n^5 + 12*n^4 + 22*n^3 + 23*n^2 - 12*n - 36)*x + (6*n^4 + 17*n^3 + 5*n^2 - 4*n + 12)*x^2 + (3*n^3 + 11*n^2 + 10*n)*x^3)*v[i - 1] + ((2*n^4 + 8*n^3 + 14*n^2 + 18*n + 12)*x - (2*n^4 + 4*n^3 + 2*n^2 + 13*n + 18)*x^2 - (4*n^3 + 10*n^2 - 2*n - 12)*x^3 - (2*n^2 + 7*n + 6)*x^4)*deriv(v[i - 1], x))/(n^5 + 4*n^4 + 5*n^3 + 11*n^2 + 18*n + 9 + (2*n^4 + 7*n^3 - n^2 - 15*n - 9)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)); 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.