Real-rooted atlas

A127773 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(144 - 156\,x + 54\,x^{2} - 6\,x^{3} - 420\,n + 394\,n x - 117\,n x^{2} + 11\,n x^{3} + 484\,n^{2} - 372\,n^{2} x + 87\,n^{2} x^{2} - 6\,n^{2} x^{3} - 285\,n^{3} + 167\,n^{3} x - 27\,n^{3} x^{2} + n^{3} x^{3} + 91\,n^{4} - 36\,n^{4} x + 3\,n^{4} x^{2} - 15\,n^{5} + 3\,n^{5} x + n^{6}\right)\,P_{n}(x) = \left(-12\,n x^{3} + 22\,n x^{4} - 12\,n x^{5} + 2\,n x^{6} + 40\,n^{2} x^{3} - 57\,n^{2} x^{4} + 24\,n^{2} x^{5} - 3\,n^{2} x^{6} - 51\,n^{3} x^{3} + 53\,n^{3} x^{4} - 15\,n^{3} x^{5} + n^{3} x^{6} + 31\,n^{4} x^{3} - 21\,n^{4} x^{4} + 3\,n^{4} x^{5} - 9\,n^{5} x^{3} + 3\,n^{5} x^{4} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(72\,n x^{2} - 126\,n x^{3} + 63\,n x^{4} - 9\,n x^{5} - 222\,n^{2} x^{2} + 294\,n^{2} x^{3} - 111\,n^{2} x^{4} + 12\,n^{2} x^{5} + 255\,n^{3} x^{2} - 237\,n^{3} x^{3} + 57\,n^{3} x^{4} - 3\,n^{3} x^{5} - 135\,n^{4} x^{2} + 78\,n^{4} x^{3} - 9\,n^{4} x^{4} + 33\,n^{5} x^{2} - 9\,n^{5} x^{3} - 3\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-216\,n x + 342\,n x^{2} - 144\,n x^{3} + 18\,n x^{4} + 522\,n^{2} x - 573\,n^{2} x^{2} + 174\,n^{2} x^{3} - 15\,n^{2} x^{4} - 465\,n^{3} x + 351\,n^{3} x^{2} - 69\,n^{3} x^{3} + 3\,n^{3} x^{4} + 195\,n^{4} x - 93\,n^{4} x^{2} + 9\,n^{4} x^{3} - 39\,n^{5} x + 9\,n^{5} x^{2} + 3\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 12 + 12\,x\)
\(P_{3}(x) = 60\,x + 30\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2121224
30603090
40018060240
5000420105525
600008401681008
70000015122521764
800000025203602880
9000000039604954455
100000000059406606600
1100000000085808589438
12000000000012012109213104
130000000000016380136517745
1400000000000021840168023520
Technical details
Node id
n-4f63621ddd1bfac6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2327141074f67bdf

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  3;
  2: 12, 12;
  3:  0, 60,  30;
  4:  0,  0, 180,  60;
  5:  0,  0,   0, 420, 105;
  6:  0,  0,   0,   0, 840,  168;
  7:  0,  0,   0,   0,   0, 1512,  252;
  8:  0,  0,   0,   0,   0,    0, 2520,  360;
  9:  0,  0,   0,   0,   0,    0,    0, 3960, 495;
%F Formula
The row polynomials satisfy (n^6-15*n^5+91*n^4-285*n^3+484*n^2-420*n+144+(3*n^5-36*n^4+167*n^3-372*n^2+394*n-156)*x+(3*n^4-27*n^3+87*n^2-117*n+54)*x^2+(n^3-6*n^2+11*n-6)*x^3)*P(n,x) = ((n^6-9*n^5+31*n^4-51*n^3+40*n^2-12*n)*x^3+(3*n^5-21*n^4+53*n^3-57*n^2+22*n)*x^4+(3*n^4-15*n^3+24*n^2-12*n)*x^5+(n^3-3*n^2+2*n)*x^6)*P(n-3,x) - ((3*n^6-33*n^5+135*n^4-255*n^3+222*n^2-72*n)*x^2+(9*n^5-78*n^4+237*n^3-294*n^2+126*n)*x^3+(9*n^4-57*n^3+111*n^2-63*n)*x^4+(3*n^3-12*n^2+9*n)*x^5)*P(n-2,x) + ((3*n^6-39*n^5+195*n^4-465*n^3+522*n^2-216*n)*x+(9*n^5-93*n^4+351*n^3-573*n^2+342*n)*x^2+(9*n^4-69*n^3+174*n^2-144*n)*x^3+(3*n^3-15*n^2+18*n)*x^4)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 12 + 12*x, P(3,x) = 60*x + 30*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 12 + 12*x;
P[3, x_] := 60*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n^6 - 9*n^5 + 31*n^4 - 51*n^3 + 40*n^2 - 12*n)*x^3 + (3*n^5 - 21*n^4 + 53*n^3 - 57*n^2 + 22*n)*x^4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x^5 + (n^3 - 3*n^2 + 2*n)*x^6)*P[n - 3, x] - ((3*n^6 - 33*n^5 + 135*n^4 - 255*n^3 + 222*n^2 - 72*n)*x^2 + (9*n^5 - 78*n^4 + 237*n^3 - 294*n^2 + 126*n)*x^3 + (9*n^4 - 57*n^3 + 111*n^2 - 63*n)*x^4 + (3*n^3 - 12*n^2 + 9*n)*x^5)*P[n - 2, x] + ((3*n^6 - 39*n^5 + 195*n^4 - 465*n^3 + 522*n^2 - 216*n)*x + (9*n^5 - 93*n^4 + 351*n^3 - 573*n^2 + 342*n)*x^2 + (9*n^4 - 69*n^3 + 174*n^2 - 144*n)*x^3 + (3*n^3 - 15*n^2 + 18*n)*x^4)*P[n - 1, x])/(n^6 - 15*n^5 + 91*n^4 - 285*n^3 + 484*n^2 - 420*n + 144 + (3*n^5 - 36*n^4 + 167*n^3 - 372*n^2 + 394*n - 156)*x + (3*n^4 - 27*n^3 + 87*n^2 - 117*n + 54)*x^2 + (n^3 - 6*n^2 + 11*n - 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 12 + 12*x; v[4] = 60*x + 30*x^2; for(n = 4, N, my(i = n + 1); v[i] = (((n^6 - 9*n^5 + 31*n^4 - 51*n^3 + 40*n^2 - 12*n)*x^3 + (3*n^5 - 21*n^4 + 53*n^3 - 57*n^2 + 22*n)*x^4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x^5 + (n^3 - 3*n^2 + 2*n)*x^6)*v[i - 3] - ((3*n^6 - 33*n^5 + 135*n^4 - 255*n^3 + 222*n^2 - 72*n)*x^2 + (9*n^5 - 78*n^4 + 237*n^3 - 294*n^2 + 126*n)*x^3 + (9*n^4 - 57*n^3 + 111*n^2 - 63*n)*x^4 + (3*n^3 - 12*n^2 + 9*n)*x^5)*v[i - 2] + ((3*n^6 - 39*n^5 + 195*n^4 - 465*n^3 + 522*n^2 - 216*n)*x + (9*n^5 - 93*n^4 + 351*n^3 - 573*n^2 + 342*n)*x^2 + (9*n^4 - 69*n^3 + 174*n^2 - 144*n)*x^3 + (3*n^3 - 15*n^2 + 18*n)*x^4)*v[i - 1])/(n^6 - 15*n^5 + 91*n^4 - 285*n^3 + 484*n^2 - 420*n + 144 + (3*n^5 - 36*n^4 + 167*n^3 - 372*n^2 + 394*n - 156)*x + (3*n^4 - 27*n^3 + 87*n^2 - 117*n + 54)*x^2 + (n^3 - 6*n^2 + 11*n - 6)*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

  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. [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.