Real-rooted atlas

A038207 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 12\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
2928946
32714810816299
481700873288251967
52433124594032806253613248
6729134683672930240950011884991903
721875678821394824673611375023184205864658715
86561235900119751318607681181775349272500783328814885276
91968396984465183401329352011179875449593292610098688510310037507185
10590493958108347738499134928099030250520664761445436321964801810357500121298076511
1117714716068628182774988609915856835945000559135368200957526408439684771305313500106481442450914078
1253144164983100950029353398442844868046085255676843744256966309667091481610417750296525005177591468816920836365141
1315943232620581644896358740255955117605385792912555213386228308537829601006544281619950790862444442001841039282180819773196183004839571
144782969105460994825071424569162277708320416998491000519262507764352835600117140953246720346849460105396391000535112578334152001261078260682251659109523566
Technical details
Node id
n-31b57962f8e1df1b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-84414087f4ed6789

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:     1;
  1:     3,      4;
  2:     9,     28,       9;
  3:    27,    148,     108,       16;
  4:    81,    700,     873,      288,       25;
  5:   243,   3124,    5940,     3280,      625,      36;
  6:   729,  13468,   36729,    30240,     9500,    1188,     49;
  7:  2187,  56788,  213948,   246736,   113750,   23184,   2058,    64;
  8:  6561, 235900, 1197513,  1860768,  1181775,  349272,  50078,  3328,   81;
  9: 19683, 969844, 6518340, 13293520, 11179875, 4495932, 926100, 98688, 5103, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(6+12*x-9*x^2)*P(n-2,x) - (6*x-12*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (5+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(6 + 12*x - 9*x^2)*P[n - 2, x] - (6*x - 12*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(6 + 12*x - 9*x^2)*v[i - 2] - (6*x - 12*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.