Real-rooted atlas

A151775 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(3 + 36\,x + 146\,x^{2} + 284\,x^{3} + 291\,x^{4} + 152\,x^{5} + 32\,x^{6} - 2\,n - 34\,n x - 162\,n x^{2} - 350\,n x^{3} - 388\,n x^{4} - 216\,n x^{5} - 48\,n x^{6} + 4\,n^{2} x + 28\,n^{2} x^{2} + 76\,n^{2} x^{3} + 100\,n^{2} x^{4} + 64\,n^{2} x^{5} + 16\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 40\,x + 228\,x^{2} + 616\,x^{3} + 914\,x^{4} + 768\,x^{5} + 344\,x^{6} + 64\,x^{7} - 8\,n x - 64\,n x^{2} - 208\,n x^{3} - 352\,n x^{4} - 328\,n x^{5} - 160\,n x^{6} - 32\,n x^{7}\right)\,P_{n-1}'(x) + \left(4\,x + 36\,x^{2} + 136\,x^{3} + 280\,x^{4} + 340\,x^{5} + 244\,x^{6} + 96\,x^{7} + 16\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
253857100
361845282727636496
4138530108165678347372250737795280
550521157872712558738420449026396709336581523156781504
62702765114545698122488631957876683961385822581916532652626782805341745348735040
719936098110991602449150729398853938210069225313175498461558281995444915717183815999230964463548318086913412096

All 15 rows

Technical details
Node id
n-51aaefce50b145f1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-50514a51a60708e3

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:             1,               3;
  2:             5,              38,               57;
  3:            61,             845,             2827,              2763;
  4:          1385,           30108,           165678,            347372,             250737;
  5:         50521,         1578727,         12558738,          42044902,           63967093,           36581523;
  6:       2702765,       114545698,       1224886319,        5787668396,        13858225819,        16532652626,          7828053417;
  7:     199360981,     10991602449,     150729398853,      938210069225,      3131754984615,      5828199544491,       5717183815999,       2309644635483;
  8:   19391512145,   1347876178808,   22928729094540,   179838785780392,    782298317080150,   2013821093811720,    3063007488423788,    2550944489331800,     898621108880097;
  9: 2404879675441, 205627071360011, 4232829305684692, 40549809708954860, 220137003799232878, 733364018069036906, 1534595693858402596, 1972903207941495932, 1427429993305584745, 445777636063460643;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -(2*n-3-(4*n^2-34*n+36)*x-(28*n^2-162*n+146)*x^2-(76*n^2-350*n+284)*x^3-(100*n^2-388*n+291)*x^4-(64*n^2-216*n+152)*x^5-(16*n^2-48*n+32)*x^6)*P(n-1,x) + (2-(8*n-40)*x-(64*n-228)*x^2-(208*n-616)*x^3-(352*n-914)*x^4-(328*n-768)*x^5-(160*n-344)*x^6-(32*n-64)*x^7)*d/dx P(n-1,x) + (4*x+36*x^2+136*x^3+280*x^4+340*x^5+244*x^6+96*x^7+16*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[(-(2*n - 3 - (4*n^2 - 34*n + 36)*x - (28*n^2 - 162*n + 146)*x^2 - (76*n^2 - 350*n + 284)*x^3 - (100*n^2 - 388*n + 291)*x^4 - (64*n^2 - 216*n + 152)*x^5 - (16*n^2 - 48*n + 32)*x^6)*P[n - 1, x] + (2 - (8*n - 40)*x - (64*n - 228)*x^2 - (208*n - 616)*x^3 - (352*n - 914)*x^4 - (328*n - 768)*x^5 - (160*n - 344)*x^6 - (32*n - 64)*x^7)*D[P[n - 1, x], x] + (4*x + 36*x^2 + 136*x^3 + 280*x^4 + 340*x^5 + 244*x^6 + 96*x^7 + 16*x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + 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] = (-(2*n - 3 - (4*n^2 - 34*n + 36)*x - (28*n^2 - 162*n + 146)*x^2 - (76*n^2 - 350*n + 284)*x^3 - (100*n^2 - 388*n + 291)*x^4 - (64*n^2 - 216*n + 152)*x^5 - (16*n^2 - 48*n + 32)*x^6)*v[i - 1] + (2 - (8*n - 40)*x - (64*n - 228)*x^2 - (208*n - 616)*x^3 - (352*n - 914)*x^4 - (328*n - 768)*x^5 - (160*n - 344)*x^6 - (32*n - 64)*x^7)*deriv(v[i - 1], x) + (4*x + 36*x^2 + 136*x^3 + 280*x^4 + 340*x^5 + 244*x^6 + 96*x^7 + 16*x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.