Real-rooted atlas

A013620 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-52 + 212\,x - 72\,x^{2} + 106\,n - 284\,n x + 72\,n x^{2} - 72\,n^{2} + 72\,n^{2} x + 18\,n^{3}\right)\,P_{n}(x) = \left(-32 + 496\,x + 486\,x^{2} - 324\,x^{3} + 68\,n - 397\,n x - 810\,n x^{2} + 324\,n x^{3} - 72\,n^{2} - 180\,n^{2} x + 324\,n^{2} x^{2} + 36\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(-64\,x - 128\,x^{2} + 900\,x^{3} - 648\,x^{4} - 72\,n x + 450\,n x^{2} - 648\,n x^{3} + 72\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
213922
36254116
423237881691
575223408103902
6222411880704772921880
76176514085329810206121088
8163841975683356641078926561664069
94198469465617962569622801180983613274
1010470422809608436960717336014762255904919531258
1125548870963203575404845419616151559101299078104980460
126123522113689613959475225096262412990780018777582531441561523447
131445888607426565100226561240956288945728784218008908138174662990722646
1433710081694085121764322560560431872059896156322106395928226925307478296915869140636
Technical details
Node id
n-38353fc84ac49a22 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-46edb3bf35b4e4c9

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     2;
  2:    13,      9;
  3:    62,     54;
  4:   232,    378,      81;
  5:   752,   2340,     810;
  6:  2224,  11880,    7047,    729;
  7:  6176,  51408,   53298,  10206;
  8: 16384, 197568,  335664, 107892,   6561;
  9: 41984, 694656, 1796256, 962280, 118098;
%F Formula
The row polynomials satisfy (18*n^3-72*n^2+106*n-52+(72*n^2-284*n+212)*x+(72*n-72)*x^2)*P(n,x) = (36*n^3-72*n^2+68*n-32+(81*n^3-180*n^2-397*n+496)*x+(324*n^2-810*n+486)*x^2+(324*n-324)*x^3)*P(n-1,x) + ((72*n^2-72*n-64)*x-(162*n^2-450*n+128)*x^2-(648*n-900)*x^3-648*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((36*n^3 - 72*n^2 + 68*n - 32 + (81*n^3 - 180*n^2 - 397*n + 496)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*P[n - 1, x] + ((72*n^2 - 72*n - 64)*x - (162*n^2 - 450*n + 128)*x^2 - (648*n - 900)*x^3 - 648*x^4)*D[P[n - 1, x], x])/(18*n^3 - 72*n^2 + 106*n - 52 + (72*n^2 - 284*n + 212)*x + (72*n - 72)*x^2)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((36*n^3 - 72*n^2 + 68*n - 32 + (81*n^3 - 180*n^2 - 397*n + 496)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*v[i - 1] + ((72*n^2 - 72*n - 64)*x - (162*n^2 - 450*n + 128)*x^2 - (648*n - 900)*x^3 - 648*x^4)*deriv(v[i - 1], x))/(18*n^3 - 72*n^2 + 106*n - 52 + (72*n^2 - 284*n + 212)*x + (72*n - 72)*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

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.