Real-rooted atlas

A059473 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n - 6\,n x - 72\,n x^{2} - 36\,n x^{3} + 54\,n x^{4} + 18\,n^{2} + 9\,n^{2} x + 54\,n^{2} x^{2} + 54\,n^{2} x^{3} + 27\,n^{3} x\right)\,P_{n}(x) = \left(-60 - 408\,x - 792\,x^{2} - 180\,x^{3} + 540\,x^{4} + 324\,x^{5} + 54\,n + 330\,n x + 630\,n x^{2} + 468\,n x^{3} + 108\,n x^{4} + 180\,n^{2} + 558\,n^{2} x + 162\,n^{2} x^{2} - 216\,n^{2} x^{3} - 54\,n^{3} - 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 48\,x + 12\,x^{2} + 612\,x^{3} + 1440\,x^{4} + 1188\,x^{5} + 324\,x^{6} + 12\,n + 144\,n x + 630\,n x^{2} + 1242\,n x^{3} + 1080\,n x^{4} + 324\,n x^{5} + 36\,n^{2} + 216\,n^{2} x + 378\,n^{2} x^{2} + 162\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110616
27212636234
3440136811522163176
4249211280168849288129641240
51346480424176184171720699847776519552
670480524088152179221723121562976505440466566403744
7360640321273611654208220872962325542413234752354585627993677630848

All 15 rows

Technical details
Node id
n-80c422646e71b8c4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ad71bebc1d95241f

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:      10,         6;
  2:      72,       126,        36;
  3:     440,      1368,      1152,        216;
  4:    2492,     11280,     16884,       9288,       1296;
  5:   13464,     80424,    176184,     171720,      69984,       7776;
  6:   70480,    524088,   1521792,    2172312,    1562976,     505440,      46656;
  7:  360640,   3212736,  11654208,   22087296,   23255424,   13234752,    3545856,    279936;
  8: 1813968,  18842112,  82113696,  194757696,  272342736,  226514880,  106515648,  24354432,   1679616;
  9: 9002528, 106863840, 544245696, 1553297472, 2720348064, 3004156512, 2061728640, 825531264, 164602368, 10077696;
%F Formula
The row polynomials satisfy (18*n^2-6*n+(27*n^3+9*n^2-6*n)*x+(54*n^2-72*n)*x^2+(54*n^2-36*n)*x^3+54*n*x^4)*P(n,x) = -(54*n^3-180*n^2-54*n+60+(108*n^3-558*n^2-330*n+408)*x-(162*n^2+630*n-792)*x^2+(216*n^2-468*n+180)*x^3-(108*n+540)*x^4-324*x^5)*P(n-1,x) + (36*n^2+12*n-8+(216*n^2+144*n-48)*x+(378*n^2+630*n+12)*x^2+(162*n^2+1242*n+612)*x^3+(1080*n+1440)*x^4+(324*n+1188)*x^5+324*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n^3 - 180*n^2 - 54*n + 60 + (108*n^3 - 558*n^2 - 330*n + 408)*x - (162*n^2 + 630*n - 792)*x^2 + (216*n^2 - 468*n + 180)*x^3 - (108*n + 540)*x^4 - 324*x^5)*P[n - 1, x] + (36*n^2 + 12*n - 8 + (216*n^2 + 144*n - 48)*x + (378*n^2 + 630*n + 12)*x^2 + (162*n^2 + 1242*n + 612)*x^3 + (1080*n + 1440)*x^4 + (324*n + 1188)*x^5 + 324*x^6)*D[P[n - 1, x], x])/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (54*n^2 - 72*n)*x^2 + (54*n^2 - 36*n)*x^3 + 54*n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(54*n^3 - 180*n^2 - 54*n + 60 + (108*n^3 - 558*n^2 - 330*n + 408)*x - (162*n^2 + 630*n - 792)*x^2 + (216*n^2 - 468*n + 180)*x^3 - (108*n + 540)*x^4 - 324*x^5)*v[i - 1] + (36*n^2 + 12*n - 8 + (216*n^2 + 144*n - 48)*x + (378*n^2 + 630*n + 12)*x^2 + (162*n^2 + 1242*n + 612)*x^3 + (1080*n + 1440)*x^4 + (324*n + 1188)*x^5 + 324*x^6)*deriv(v[i - 1], x))/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (54*n^2 - 72*n)*x^2 + (54*n^2 - 36*n)*x^3 + 54*n*x^4)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.