Real-rooted atlas

A125092 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 5\,x + 10\,x^{2} + 10\,x^{3} + 5\,x^{4} + x^{5}\right)\,P_{n-5}(x) + \left(-5 - 20\,x - 30\,x^{2} - 20\,x^{3} - 5\,x^{4}\right)\,P_{n-4}(x) + \left(10 + 30\,x + 30\,x^{2} + 10\,x^{3}\right)\,P_{n-3}(x) + \left(-10 - 20\,x - 10\,x^{2}\right)\,P_{n-2}(x) + \left(5 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 8\,x\)
\(P_{2}(x) = 9 + 52\,x + 27\,x^{2}\)
\(P_{3}(x) = 13 + 132\,x + 225\,x^{2} + 64\,x^{3}\)
\(P_{4}(x) = 17 + 248\,x + 738\,x^{2} + 656\,x^{3} + 125\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15813
29522788
31313222564434
4172487386561251784
5214001710264015252166512
625588328572807275306034321856
7298125607162402327516884553751268896

All 15 rows

Technical details
Node id
n-d9a2fda52d5e8d86 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-fbd5d0119ffa01e5

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,    8;
  2:  9,   52,    27;
  3: 13,  132,   225,    64;
  4: 17,  248,   738,   656,    125;
  5: 21,  400,  1710,  2640,   1525,    216;
  6: 25,  588,  3285,  7280,   7275,   3060,    343;
  7: 29,  812,  5607, 16240,  23275,  16884,   5537,   512;
  8: 33, 1072,  8820, 31584,  59150,  61488,  34692,  9280,   729;
  9: 37, 1368, 13068, 55776, 129150, 175392, 141708, 65088, 14661, 1000;
%F Formula
The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 9 + 52*x + 27*x^2, P(3,x) = 13 + 132*x + 225*x^2 + 64*x^3, P(4,x) = 17 + 248*x + 738*x^2 + 656*x^3 + 125*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 8*x;
P[2, x_] := 9 + 52*x + 27*x^2;
P[3, x_] := 13 + 132*x + 225*x^2 + 64*x^3;
P[4, x_] := 17 + 248*x + 738*x^2 + 656*x^3 + 125*x^4;
P[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 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 + 8*x; v[3] = 9 + 52*x + 27*x^2; v[4] = 13 + 132*x + 225*x^2 + 64*x^3; v[5] = 17 + 248*x + 738*x^2 + 656*x^3 + 125*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 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.