Real-rooted atlas

A130534 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-81 + 36\,x + 9\,x^{2} + 108\,n - 24\,n x - 54\,n^{2} + 4\,n^{2} x + 12\,n^{3} - n^{4}\right)\,P_{n-4}(x) + \left(-36\,x^{2} + 4\,x^{3} + 24\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-4}'(x) + \left(-65 + 20\,x + 76\,n - 8\,n x - 30\,n^{2} + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-20\,x^{2} + 8\,n x^{2}\right)\,P_{n-3}'(x) + \left(-25 + 4\,x + 24\,n - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2}\right)\,P_{n-2}'(x) + \left(-6 + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)
\(P_{3}(x) = 6 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2246
362430
4241409173
51209001351155
672064961575168807
75040525281764044875656
840320472496202041873625723618
93628804690800242392515120011257629930
10362880051014304307523702524368330003687952878
11399168006036719044139615704220092881675023761100570328
124790016007726238208591486152471984369618736575978124914858779464
136227020800106386868224896193338041265509044841674132532792764459215308338336
148717829120015686271912961437494451120230549166848921029037598990892245245643333158627040
Technical details
Node id
n-f0a112d8bb68d045 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-72fed22386e65f36

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:      2,       4;
  3:      6,      24;
  4:     24,     140,       9;
  5:    120,     900,     135;
  6:    720,    6496,    1575,     16;
  7:   5040,   52528,   17640,    448;
  8:  40320,  472496,  202041,   8736,   25;
  9: 362880, 4690800, 2423925, 151200, 1125;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81-(4*n^2-24*n+36)*x-9*x^2)*P(n-4,x) - ((4*n^2-24*n+36)*x^2-4*x^3)*d/dx P(n-4,x) + (4*n^3-30*n^2+76*n-65-(8*n-20)*x)*P(n-3,x) + (8*n-20)*x^2*d/dx P(n-3,x) - (6*n^2-24*n+25-4*x)*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (4*n-6)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + 4*x, P(3,x) = 6 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[3, x_] := 6 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (4*n^2 - 24*n + 36)*x - 9*x^2)*P[n - 4, x] - ((4*n^2 - 24*n + 36)*x^2 - 4*x^3)*D[P[n - 4, x], x] + (4*n^3 - 30*n^2 + 76*n - 65 - (8*n - 20)*x)*P[n - 3, x] + (8*n - 20)*x^2*D[P[n - 3, x], x] - (6*n^2 - 24*n + 25 - 4*x)*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (4*n - 6)*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; v[3] = 2 + 4*x; v[4] = 6 + 24*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (4*n^2 - 24*n + 36)*x - 9*x^2)*v[i - 4] - ((4*n^2 - 24*n + 36)*x^2 - 4*x^3)*deriv(v[i - 4], x) + (4*n^3 - 30*n^2 + 76*n - 65 - (8*n - 20)*x)*v[i - 3] + (8*n - 20)*x^2*deriv(v[i - 3], x) - (6*n^2 - 24*n + 25 - 4*x)*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (4*n - 6)*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

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.