Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,n - 14\,n^{2} + 12\,n^{3} + 7\,n^{4} - 6\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(24 + 72\,x + 72\,x^{2} + 24\,x^{3} - 8\,n^{2} - 24\,n^{2} x - 24\,n^{2} x^{2} - 8\,n^{2} x^{3} + 15\,n^{3} + 45\,n^{3} x + 45\,n^{3} x^{2} + 15\,n^{3} x^{3} - 5\,n^{4} - 15\,n^{4} x - 15\,n^{4} x^{2} - 5\,n^{4} x^{3} - 3\,n^{5} - 9\,n^{5} x - 9\,n^{5} x^{2} - 3\,n^{5} x^{3} + n^{6} + 3\,n^{6} x + 3\,n^{6} x^{2} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(12 + 48\,x + 72\,x^{2} + 48\,x^{3} + 12\,x^{4} + 6\,n + 24\,n x + 36\,n x^{2} + 24\,n x^{3} + 6\,n x^{4} - 6\,n^{2} - 24\,n^{2} x - 36\,n^{2} x^{2} - 24\,n^{2} x^{3} - 6\,n^{2} x^{4}\right)\,P_{n-3}'(x) + \left(36\,n + 72\,n x + 36\,n x^{2} + 6\,n^{2} + 12\,n^{2} x + 6\,n^{2} x^{2} - 60\,n^{3} - 120\,n^{3} x - 60\,n^{3} x^{2} + 9\,n^{4} + 18\,n^{4} x + 9\,n^{4} x^{2} + 12\,n^{5} + 24\,n^{5} x + 12\,n^{5} x^{2} - 3\,n^{6} - 6\,n^{6} x - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-24 - 72\,x - 72\,x^{2} - 24\,x^{3} - 24\,n - 72\,n x - 72\,n x^{2} - 24\,n x^{3} + 12\,n^{2} + 36\,n^{2} x + 36\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-72\,n - 72\,n x - 30\,n^{2} - 30\,n^{2} x + 63\,n^{3} + 63\,n^{3} x + 3\,n^{4} + 3\,n^{4} x - 15\,n^{5} - 15\,n^{5} x + 3\,n^{6} + 3\,n^{6} x\right)\,P_{n-1}(x) + \left(18\,n + 36\,n x + 18\,n x^{2} - 6\,n^{2} - 12\,n^{2} x - 6\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
\(P_{2}(x) = 3 + 16\,x + 9\,x^{2}\)
\(P_{3}(x) = 4 + 36\,x + 54\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2316928
34365416110
456416212825384
56100360480250361232
6714467512801125432493712
7819611342800350022686866410656

All 15 rows

Technical details
Node id
n-48ef4a5e7cd19d2a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-329289f0f6a2a0c6

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:  2,   4;
  2:  3,  16,    9;
  3:  4,  36,   54,   16;
  4:  5,  64,  162,  128,    25;
  5:  6, 100,  360,  480,   250,    36;
  6:  7, 144,  675, 1280,  1125,   432,    49;
  7:  8, 196, 1134, 2800,  3500,  2268,   686,   64;
  8:  9, 256, 1764, 5376,  8750,  8064,  4116, 1024,   81;
  9: 10, 324, 2592, 9408, 18900, 22680, 16464, 6912, 1458, 100;
%F Formula
The row polynomials satisfy (n^6-6*n^5+7*n^4+12*n^3-14*n^2-12*n)*P(n,x) = (n^6-3*n^5-5*n^4+15*n^3-8*n^2+24+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x+(3*n^6-9*n^5-15*n^4+45*n^3-24*n^2+72)*x^2+(n^6-3*n^5-5*n^4+15*n^3-8*n^2+24)*x^3)*P(n-3,x) - (6*n^2-6*n-12+(24*n^2-24*n-48)*x+(36*n^2-36*n-72)*x^2+(24*n^2-24*n-48)*x^3+(6*n^2-6*n-12)*x^4)*d/dx P(n-3,x) - (3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n+(6*n^6-24*n^5-18*n^4+120*n^3-12*n^2-72*n)*x+(3*n^6-12*n^5-9*n^4+60*n^3-6*n^2-36*n)*x^2)*P(n-2,x) + (12*n^2-24*n-24+(36*n^2-72*n-72)*x+(36*n^2-72*n-72)*x^2+(12*n^2-24*n-24)*x^3)*d/dx P(n-2,x) + (3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n+(3*n^6-15*n^5+3*n^4+63*n^3-30*n^2-72*n)*x)*P(n-1,x) - (6*n^2-18*n+(12*n^2-36*n)*x+(6*n^2-18*n)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 3 + 16*x + 9*x^2, P(3,x) = 4 + 36*x + 54*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 3 + 16*x + 9*x^2;
P[3, x_] := 4 + 36*x + 54*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*P[n - 3, x] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*D[P[n - 3, x], x] - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*P[n - 2, x] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*D[P[n - 2, x], x] + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*P[n - 1, x] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*D[P[n - 1, x], x])/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)]];
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] = 2 + 4*x; v[3] = 3 + 16*x + 9*x^2; v[4] = 4 + 36*x + 54*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 24*n^2 + 72)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 8*n^2 + 24)*x^3)*v[i - 3] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*deriv(v[i - 3], x) - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 12*n^2 - 72*n)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 6*n^2 - 36*n)*x^2)*v[i - 2] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*deriv(v[i - 2], x) + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 30*n^2 - 72*n)*x)*v[i - 1] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*deriv(v[i - 1], x))/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)); 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

Candidate equalities ?

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.