Real-rooted atlas

A124932 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(32 + 80\,x + 80\,x^{2} + 40\,x^{3} + 10\,x^{4} + x^{5}\right)\,P_{n-5}(x) + \left(-80 - 160\,x - 120\,x^{2} - 40\,x^{3} - 5\,x^{4}\right)\,P_{n-4}(x) + \left(80 + 120\,x + 60\,x^{2} + 10\,x^{3}\right)\,P_{n-3}(x) + \left(-40 - 40\,x - 10\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 12\,x\)
\(P_{2}(x) = 18 + 84\,x + 54\,x^{2}\)
\(P_{3}(x) = 56 + 384\,x + 486\,x^{2} + 160\,x^{3}\)
\(P_{4}(x) = 160 + 1440\,x + 2700\,x^{2} + 1760\,x^{3} + 375\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151217
2188454156
3563844861601086
41601440270017603756435
543248001188011520487575634263
6112014784453605824036750113401372168966
728164300815724825088021000096768233242304786348

All 15 rows

Technical details
Node id
n-97efd101204ca497 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6c80aa0fd6d7c8cd

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:     5,     12;
  2:    18,     84,      54;
  3:    56,    384,     486,     160;
  4:   160,   1440,    2700,    1760,     375;
  5:   432,   4800,   11880,   11520,    4875,     756;
  6:  1120,  14784,   45360,   58240,   36750,   11340,    1372;
  7:  2816,  43008,  157248,  250880,  210000,   96768,   23324,   2304;
  8:  6912, 119808,  508032,  967680, 1008000,  616896,  222264,  43776,  3645;
  9: 16640, 322560, 1555200, 3440640, 4284000, 3265920, 1564080, 460800, 76545, 5500;
%F Formula
The row polynomials satisfy P(n,x) = (32+80*x+80*x^2+40*x^3+10*x^4+x^5)*P(n-5,x) - (80+160*x+120*x^2+40*x^3+5*x^4)*P(n-4,x) + (80+120*x+60*x^2+10*x^3)*P(n-3,x) - (40+40*x+10*x^2)*P(n-2,x) + (10+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 5 + 12*x, P(2,x) = 18 + 84*x + 54*x^2, P(3,x) = 56 + 384*x + 486*x^2 + 160*x^3, P(4,x) = 160 + 1440*x + 2700*x^2 + 1760*x^3 + 375*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 12*x;
P[2, x_] := 18 + 84*x + 54*x^2;
P[3, x_] := 56 + 384*x + 486*x^2 + 160*x^3;
P[4, x_] := 160 + 1440*x + 2700*x^2 + 1760*x^3 + 375*x^4;
P[n_, x_] := P[n, x] = Expand[(32 + 80*x + 80*x^2 + 40*x^3 + 10*x^4 + x^5)*P[n - 5, x] - (80 + 160*x + 120*x^2 + 40*x^3 + 5*x^4)*P[n - 4, x] + (80 + 120*x + 60*x^2 + 10*x^3)*P[n - 3, x] - (40 + 40*x + 10*x^2)*P[n - 2, x] + (10 + 5*x)*P[n - 1, 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] = 5 + 12*x; v[3] = 18 + 84*x + 54*x^2; v[4] = 56 + 384*x + 486*x^2 + 160*x^3; v[5] = 160 + 1440*x + 2700*x^2 + 1760*x^3 + 375*x^4; for(n = 5, N, my(i = n + 1); v[i] = (32 + 80*x + 80*x^2 + 40*x^3 + 10*x^4 + x^5)*v[i - 5] - (80 + 160*x + 120*x^2 + 40*x^3 + 5*x^4)*v[i - 4] + (80 + 120*x + 60*x^2 + 10*x^3)*v[i - 3] - (40 + 40*x + 10*x^2)*v[i - 2] + (10 + 5*x)*v[i - 1]); 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.