Real-rooted atlas

A094305 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 - 12\,x + 4\,x^{2} + 12\,n x - 12\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(64 + 96\,x + 32\,x^{2} - 16\,x^{3} - 12\,x^{4} - 2\,x^{5} + 96\,n x + 144\,n x^{2} + 64\,n x^{3} - 6\,n x^{5} - n x^{6} + 48\,n^{2} x^{2} + 72\,n^{2} x^{3} + 36\,n^{2} x^{4} + 6\,n^{2} x^{5} + 8\,n^{3} x^{3} + 12\,n^{3} x^{4} + 6\,n^{3} x^{5} + n^{3} x^{6}\right)\,P_{n-3}(x) + \left(-96 - 48\,x + 72\,x^{2} + 60\,x^{3} + 12\,x^{4} - 144\,n x - 96\,n x^{2} + 36\,n x^{3} + 36\,n x^{4} + 6\,n x^{5} - 72\,n^{2} x^{2} - 60\,n^{2} x^{3} - 6\,n^{2} x^{4} + 3\,n^{2} x^{5} - 12\,n^{3} x^{3} - 12\,n^{3} x^{4} - 3\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(48 - 24\,x - 36\,x^{2} + 6\,x^{3} + 6\,x^{4} + 72\,n x - 12\,n x^{2} - 30\,n x^{3} - 3\,n x^{4} + 36\,n^{2} x^{2} + 6\,n^{2} x^{3} - 6\,n^{2} x^{4} + 6\,n^{3} x^{3} + 3\,n^{3} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)
\(P_{2}(x) = 24 + 48\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
224481890
38024018040540
42409601080480752835
5672336050403360105012613608
617921075220160179208400201619661236
7460832256725768064050400181443528288262440

All 15 rows

Technical details
Node id
n-fbe3a8180125f580 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     6,      6;
  2:    24,     48,     18;
  3:    80,    240,    180,      40;
  4:   240,    960,   1080,     480,      75;
  5:   672,   3360,   5040,    3360,    1050,    126;
  6:  1792,  10752,  20160,   17920,    8400,   2016,    196;
  7:  4608,  32256,  72576,   80640,   50400,  18144,   3528,   288;
  8: 11520,  92160, 241920,  322560,  252000, 120960,  35280,  5760,  405;
  9: 28160, 253440, 760320, 1182720, 1108800, 665280, 258720, 63360, 8910, 550;
%F Formula
The row polynomials satisfy (8+(12*n-12)*x+(6*n^2-12*n+4)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n,x) = (64+(96*n+96)*x+(48*n^2+144*n+32)*x^2+(8*n^3+72*n^2+64*n-16)*x^3+(12*n^3+36*n^2-12)*x^4+(6*n^3+6*n^2-6*n-2)*x^5+(n^3-n)*x^6)*P(n-3,x) - (96+(144*n+48)*x+(72*n^2+96*n-72)*x^2+(12*n^3+60*n^2-36*n-60)*x^3+(12*n^3+6*n^2-36*n-12)*x^4+(3*n^3-3*n^2-6*n)*x^5)*P(n-2,x) + (48+(72*n-24)*x+(36*n^2-12*n-36)*x^2+(6*n^3+6*n^2-30*n+6)*x^3+(3*n^3-6*n^2-3*n+6)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 6*x, P(2,x) = 24 + 48*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[2, x_] := 24 + 48*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (96*n + 96)*x + (48*n^2 + 144*n + 32)*x^2 + (8*n^3 + 72*n^2 + 64*n - 16)*x^3 + (12*n^3 + 36*n^2 - 12)*x^4 + (6*n^3 + 6*n^2 - 6*n - 2)*x^5 + (n^3 - n)*x^6)*P[n - 3, x] - (96 + (144*n + 48)*x + (72*n^2 + 96*n - 72)*x^2 + (12*n^3 + 60*n^2 - 36*n - 60)*x^3 + (12*n^3 + 6*n^2 - 36*n - 12)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*P[n - 2, x] + (48 + (72*n - 24)*x + (36*n^2 - 12*n - 36)*x^2 + (6*n^3 + 6*n^2 - 30*n + 6)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*P[n - 1, x])/(8 + (12*n - 12)*x + (6*n^2 - 12*n + 4)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)]];
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] = 6 + 6*x; v[3] = 24 + 48*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((64 + (96*n + 96)*x + (48*n^2 + 144*n + 32)*x^2 + (8*n^3 + 72*n^2 + 64*n - 16)*x^3 + (12*n^3 + 36*n^2 - 12)*x^4 + (6*n^3 + 6*n^2 - 6*n - 2)*x^5 + (n^3 - n)*x^6)*v[i - 3] - (96 + (144*n + 48)*x + (72*n^2 + 96*n - 72)*x^2 + (12*n^3 + 60*n^2 - 36*n - 60)*x^3 + (12*n^3 + 6*n^2 - 36*n - 12)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*v[i - 2] + (48 + (72*n - 24)*x + (36*n^2 - 12*n - 36)*x^2 + (6*n^3 + 6*n^2 - 30*n + 6)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*v[i - 1])/(8 + (12*n - 12)*x + (6*n^2 - 12*n + 4)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)); 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

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  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.