Real-rooted atlas

A003506 → (θ+1)² → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 14\,x - 2\,x^{2} + 4\,n - 17\,n x + 5\,n x^{2} + 5\,n^{2} x - 4\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(4\,n + 2\,n x - 2\,n x^{2} + 5\,n^{2} x + 5\,n^{2} x^{2} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
288
3245478
448216192456
5805409605002080
612010802880300010808160
716818906720105007560205828896
8224302413440280003024016464358494976
9288453624192630009072074088322565832294912
10360648040320126000226800246960161280583209000875520
1144089106336023100049896067914059136032076099000133102506240
125281188095040396000997920162993617740801283040594000159720190086961152
13624154441372806435001853280353152846126084169880257400010381802471042636418849792
147281965619219210010003243240706305610762752116756649009000484484017297283690963567249946624
Technical details
Node id
n-83241ad01b113359 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-da7c9111cf4f46ae

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:   0;
  2:   8;
  3:  24,   54;
  4:  48,  216,   192;
  5:  80,  540,   960,   500;
  6: 120, 1080,  2880,  3000,  1080;
  7: 168, 1890,  6720, 10500,  7560,  2058;
  8: 224, 3024, 13440, 28000, 30240, 16464,  3584;
  9: 288, 4536, 24192, 63000, 90720, 74088, 32256, 5832;
%F Formula
The row polynomials satisfy (4*n-8+(5*n^2-17*n+14)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n,x) = (4*n+(5*n^2+2*n)*x+(n^3+5*n^2-2*n)*x^2+n^3*x^3)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (5*n^2 + 2*n)*x + (n^3 + 5*n^2 - 2*n)*x^2 + n^3*x^3)*P[n - 1, x])/(4*n - 8 + (5*n^2 - 17*n + 14)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)]];
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] = 0; v[3] = 8; for(n = 3, N, my(i = n + 1); v[i] = ((4*n + (5*n^2 + 2*n)*x + (n^3 + 5*n^2 - 2*n)*x^2 + n^3*x^3)*v[i - 1])/(4*n - 8 + (5*n^2 - 17*n + 14)*x + (n^3 - 4*n^2 + 5*n - 2)*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.