Real-rooted atlas

A125092 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 15\,x^{2} - 20\,x^{3} - 15\,x^{4} - 6\,x^{5} - x^{6}\right)\,P_{n-6}(x) + \left(6 + 30\,x + 60\,x^{2} + 60\,x^{3} + 30\,x^{4} + 6\,x^{5}\right)\,P_{n-5}(x) + \left(-15 - 60\,x - 90\,x^{2} - 60\,x^{3} - 15\,x^{4}\right)\,P_{n-4}(x) + \left(20 + 60\,x + 60\,x^{2} + 20\,x^{3}\right)\,P_{n-3}(x) + \left(-15 - 30\,x - 15\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 32\,x\)
\(P_{2}(x) = 1 + 64\,x + 243\,x^{2}\)
\(P_{3}(x) = 1 + 96\,x + 729\,x^{2} + 1024\,x^{3}\)
\(P_{4}(x) = 1 + 128\,x + 1458\,x^{2} + 4096\,x^{3} + 3125\,x^{4}\)
\(P_{5}(x) = 1 + 160\,x + 2430\,x^{2} + 10240\,x^{3} + 15625\,x^{4} + 7776\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113233
2164243308
319672910241850
411281458409631258808
5116024301024015625777636232
61192364520480468754665616807134656
7122451033584010937516329611764932768464256
81256680457344218750435456470596262144590491510400
91288874886016393750979776141178811796485314411000004691456
101320109351228806562501959552352947039321602657205100000016105114029824
111352133651689601031250359251277648341081344097430855500000177156124883240648192
121384160382252801546875615859215529668259522562922925522000000106293662985984371293114644992
131416189542928642234375100077122884081256229888759960637150000046060586194088964826809537824315955200
141448221133727363128125155675525047142111245977617732414720020000016121205190574848337876637529536759375853409792
Technical details
Node id
n-2f512b7fc0ea0103 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-41a7dc442fee632f

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: 1,  32;
  2: 1,  64,  243;
  3: 1,  96,  729,  1024;
  4: 1, 128, 1458,  4096,   3125;
  5: 1, 160, 2430, 10240,  15625,   7776;
  6: 1, 192, 3645, 20480,  46875,  46656,   16807;
  7: 1, 224, 5103, 35840, 109375, 163296,  117649,   32768;
  8: 1, 256, 6804, 57344, 218750, 435456,  470596,  262144,  59049;
  9: 1, 288, 8748, 86016, 393750, 979776, 1411788, 1179648, 531441, 100000;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+15*x^2+20*x^3+15*x^4+6*x^5+x^6)*P(n-6,x) + (6+30*x+60*x^2+60*x^3+30*x^4+6*x^5)*P(n-5,x) - (15+60*x+90*x^2+60*x^3+15*x^4)*P(n-4,x) + (20+60*x+60*x^2+20*x^3)*P(n-3,x) - (15+30*x+15*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1 + 32*x, P(2,x) = 1 + 64*x + 243*x^2, P(3,x) = 1 + 96*x + 729*x^2 + 1024*x^3, P(4,x) = 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4, P(5,x) = 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 32*x;
P[2, x_] := 1 + 64*x + 243*x^2;
P[3, x_] := 1 + 96*x + 729*x^2 + 1024*x^3;
P[4, x_] := 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4;
P[5, x_] := 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*P[n - 6, x] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*P[n - 5, x] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*P[n - 4, x] + (20 + 60*x + 60*x^2 + 20*x^3)*P[n - 3, x] - (15 + 30*x + 15*x^2)*P[n - 2, x] + (6 + 6*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] = 1 + 32*x; v[3] = 1 + 64*x + 243*x^2; v[4] = 1 + 96*x + 729*x^2 + 1024*x^3; v[5] = 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4; v[6] = 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5; for(n = 6, N, my(i = n + 1); v[i] = -(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*v[i - 6] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*v[i - 5] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*v[i - 4] + (20 + 60*x + 60*x^2 + 20*x^3)*v[i - 3] - (15 + 30*x + 15*x^2)*v[i - 2] + (6 + 6*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.