Real-rooted atlas

A352880 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 7\,n + 3\,n x + 5\,n^{2} + 10\,n^{2} x + n^{3} + 12\,n^{3} x + 6\,n^{4} x + n^{5} x\right)\,P_{n}(x) = \left(12 + 16\,n + 32\,n x + 6\,n x^{2} + 7\,n^{2} + 50\,n^{2} x + 31\,n^{2} x^{2} + n^{3} + 29\,n^{3} x + 38\,n^{3} x^{2} + 8\,n^{4} x + 18\,n^{4} x^{2} + n^{5} x + 3\,n^{5} x^{2}\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{3} + 7\,n x + 16\,n x^{2} - 23\,n x^{3} + 2\,n^{2} x + 24\,n^{2} x^{2} - 26\,n^{2} x^{3} + 12\,n^{3} x^{2} - 12\,n^{3} x^{3} + 2\,n^{4} x^{2} - 2\,n^{4} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
088
1182442
23212848208
35040045080980
472960216011521204464
598196073507840245016819866
61283584201603584022400460822486944
7162604847628127008132300544327938288375804

All 15 rows

Technical details
Node id
n-31c316014331006b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b29d29aac1e7588a

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:  18,    24;
  2:  32,   128,     48;
  3:  50,   400,    450,     80;
  4:  72,   960,   2160,   1152,     120;
  5:  98,  1960,   7350,   7840,    2450,     168;
  6: 128,  3584,  20160,  35840,   22400,    4608,     224;
  7: 162,  6048,  47628, 127008,  132300,   54432,    7938,    288;
  8: 200,  9600, 100800, 376320,  588000,  403200,  117600,  12800,   360;
  9: 242, 14520, 196020, 975744, 2134440, 2195424, 1067220, 232320, 19602, 440;
%F Formula
The row polynomials satisfy (n^3+5*n^2+7*n+3+(n^5+6*n^4+12*n^3+10*n^2+3*n)*x)*P(n,x) = (n^3+7*n^2+16*n+12+(n^5+8*n^4+29*n^3+50*n^2+32*n)*x+(3*n^5+18*n^4+38*n^3+31*n^2+6*n)*x^2)*P(n-1,x) + ((2*n^2+7*n+6)*x+(2*n^4+12*n^3+24*n^2+16*n)*x^2-(2*n^4+12*n^3+26*n^2+23*n+6)*x^3)*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^3 + 7*n^2 + 16*n + 12 + (n^5 + 8*n^4 + 29*n^3 + 50*n^2 + 32*n)*x + (3*n^5 + 18*n^4 + 38*n^3 + 31*n^2 + 6*n)*x^2)*P[n - 1, x] + ((2*n^2 + 7*n + 6)*x + (2*n^4 + 12*n^3 + 24*n^2 + 16*n)*x^2 - (2*n^4 + 12*n^3 + 26*n^2 + 23*n + 6)*x^3)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 7*n + 3 + (n^5 + 6*n^4 + 12*n^3 + 10*n^2 + 3*n)*x)]];
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^3 + 7*n^2 + 16*n + 12 + (n^5 + 8*n^4 + 29*n^3 + 50*n^2 + 32*n)*x + (3*n^5 + 18*n^4 + 38*n^3 + 31*n^2 + 6*n)*x^2)*v[i - 1] + ((2*n^2 + 7*n + 6)*x + (2*n^4 + 12*n^3 + 24*n^2 + 16*n)*x^2 - (2*n^4 + 12*n^3 + 26*n^2 + 23*n + 6)*x^3)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 7*n + 3 + (n^5 + 6*n^4 + 12*n^3 + 10*n^2 + 3*n)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.