Real-rooted atlas

A056939 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(12 + 36\,x + 36\,x^{2} + 12\,x^{3} + 7\,n + 21\,n x + 21\,n x^{2} + 7\,n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n}(x) = \left(60 + 396\,x + 1116\,x^{2} + 1716\,x^{3} + 1512\,x^{4} + 720\,x^{5} + 144\,x^{6} - 55\,n - 459\,n x - 1509\,n x^{2} - 2539\,n x^{3} - 2328\,n x^{4} - 1110\,n x^{5} - 216\,n x^{6} + 35\,n^{2} + 243\,n^{2} x + 693\,n^{2} x^{2} + 1043\,n^{2} x^{3} + 876\,n^{2} x^{4} + 390\,n^{2} x^{5} + 72\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(72 + 624\,x + 2304\,x^{2} + 4704\,x^{3} + 5736\,x^{4} + 4176\,x^{5} + 1680\,x^{6} + 288\,x^{7} - 36\,n - 327\,n x - 1242\,n x^{2} - 2562\,n x^{3} - 3108\,n x^{4} - 2223\,n x^{5} - 870\,n x^{6} - 144\,n x^{7}\right)\,P_{n-1}'(x) + \left(12 + 126\,x + 570\,x^{2} + 1452\,x^{3} + 2280\,x^{4} + 2262\,x^{5} + 1386\,x^{6} + 480\,x^{7} + 72\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
26181337
3229914569335
492552122011764173457
542231659360136359780276339125
6207418666691651350001463458352619609474385
7107541129175029291231335178929115429337310031467936067955

All 15 rows

Technical details
Node id
n-e4b4d28eba73c03c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-40b4fbdb5cd978eb

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,       3;
  2:      6,      18,       13;
  3:     22,      99,      145,       69;
  4:     92,     552,     1220,     1176,       417;
  5:    422,    3165,     9360,    13635,      9780,      2763;
  6:   2074,   18666,    69165,   135000,    146345,     83526,     19609;
  7:  10754,  112917,   502929,  1231335,   1789291,   1542933,    731003,    146793;
  8:  58202,  698424,  3633812, 10703952,  19520606,  22565424,  16145612,   6537264,  1146833;
  9: 326240, 4404240, 26214060, 90269550, 198160410, 287539560, 275767254, 168552738, 59576718, 9278595;
%F Formula
The row polynomials satisfy (n^2+7*n+12+(3*n^2+21*n+36)*x+(3*n^2+21*n+36)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (35*n^2-55*n+60+(243*n^2-459*n+396)*x+(693*n^2-1509*n+1116)*x^2+(1043*n^2-2539*n+1716)*x^3+(876*n^2-2328*n+1512)*x^4+(390*n^2-1110*n+720)*x^5+(72*n^2-216*n+144)*x^6)*P(n-1,x) - (36*n-72+(327*n-624)*x+(1242*n-2304)*x^2+(2562*n-4704)*x^3+(3108*n-5736)*x^4+(2223*n-4176)*x^5+(870*n-1680)*x^6+(144*n-288)*x^7)*d/dx P(n-1,x) + (12+126*x+570*x^2+1452*x^3+2280*x^4+2262*x^5+1386*x^6+480*x^7+72*x^8)*d^2/dx^2 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[((35*n^2 - 55*n + 60 + (243*n^2 - 459*n + 396)*x + (693*n^2 - 1509*n + 1116)*x^2 + (1043*n^2 - 2539*n + 1716)*x^3 + (876*n^2 - 2328*n + 1512)*x^4 + (390*n^2 - 1110*n + 720)*x^5 + (72*n^2 - 216*n + 144)*x^6)*P[n - 1, x] - (36*n - 72 + (327*n - 624)*x + (1242*n - 2304)*x^2 + (2562*n - 4704)*x^3 + (3108*n - 5736)*x^4 + (2223*n - 4176)*x^5 + (870*n - 1680)*x^6 + (144*n - 288)*x^7)*D[P[n - 1, x], x] + (12 + 126*x + 570*x^2 + 1452*x^3 + 2280*x^4 + 2262*x^5 + 1386*x^6 + 480*x^7 + 72*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*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; for(n = 1, N, my(i = n + 1); v[i] = ((35*n^2 - 55*n + 60 + (243*n^2 - 459*n + 396)*x + (693*n^2 - 1509*n + 1116)*x^2 + (1043*n^2 - 2539*n + 1716)*x^3 + (876*n^2 - 2328*n + 1512)*x^4 + (390*n^2 - 1110*n + 720)*x^5 + (72*n^2 - 216*n + 144)*x^6)*v[i - 1] - (36*n - 72 + (327*n - 624)*x + (1242*n - 2304)*x^2 + (2562*n - 4704)*x^3 + (3108*n - 5736)*x^4 + (2223*n - 4176)*x^5 + (870*n - 1680)*x^6 + (144*n - 288)*x^7)*deriv(v[i - 1], x) + (12 + 126*x + 570*x^2 + 1452*x^3 + 2280*x^4 + 2262*x^5 + 1386*x^6 + 480*x^7 + 72*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023