Real-rooted atlas

A013610 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 3\,x + n - 6\,n x + 3\,n^{2} x\right)\,P_{n}(x) = \left(n + 3\,n x + 3\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
263642
39108243360
41221697212962496
51536024306480607515360
618540486019440364502624487552
721756850545360127575183708107163473088
824100813608907203402007348328573044199042457600
927129620412163296765450220449638578683779136159432312386304
103016202916027216015309005511240128595601889568015943230590490060948480
11331980400954276802806650121247283536379069284160876877656495390021434787294125568
123623765346064152048114002424945684873096207852480350751060389723400257217444765275041396703232
13392808694989266407818525450347041838917085404164481139940945168880140016719133869948575522694405876543114240
14423276884521297296121621507881073236778341612609717123191834646591080490078022624686964002864377216821893746192430299652096
Technical details
Node id
n-136eeb95ca41b995 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3151c955f16eaf46

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  3;
  2:  6,   36;
  3:  9,  108,   243;
  4: 12,  216,   972,   1296;
  5: 15,  360,  2430,   6480,   6075;
  6: 18,  540,  4860,  19440,  36450,   26244;
  7: 21,  756,  8505,  45360, 127575,  183708,  107163;
  8: 24, 1008, 13608,  90720, 340200,  734832,  857304,  419904;
  9: 27, 1296, 20412, 163296, 765450, 2204496, 3857868, 3779136, 1594323;
%F Formula
The row polynomials satisfy (n-1+(3*n^2-6*n+3)*x)*P(n,x) = (n+(3*n^2+3*n)*x+9*n^2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n + (3*n^2 + 3*n)*x + 9*n^2*x^2)*P[n - 1, x])/(n - 1 + (3*n^2 - 6*n + 3)*x)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n + (3*n^2 + 3*n)*x + 9*n^2*x^2)*v[i - 1])/(n - 1 + (3*n^2 - 6*n + 3)*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.