Real-rooted atlas

mobius_x_over_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 27\,x^{2} + 37\,x^{3} + 24\,x^{4} + 6\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 14\,x^{3} + 16\,x^{4} + 9\,x^{5} + 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11449
2110344824117
31201324166885761922025
41384142192661612000129607680192043821
51721200952044040129312249184314880251520115200230401137969
61138337837792246360104011229961766026496848198481984005187840193536032256034476597
7126894361435841256080715052828254016802539521666670082538547202806917122193408001148313603612672051609601193741145

All 15 rows

Technical details
Node id
n-3e49d29390a1ef28 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d0b4c8550cf1eaf3

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,    4,     4;
  2: 1,   10,    34,      48,       24;
  3: 1,   20,   132,     416,      688,       576,        192;
  4: 1,   38,   414,    2192,     6616,     12000,      12960,       7680,        1920;
  5: 1,   72,  1200,    9520,    44040,    129312,     249184,     314880,      251520,      115200,        23040;
  6: 1,  138,  3378,   37792,   246360,   1040112,    2996176,    6026496,     8481984,     8198400,      5187840,      1935360,       322560;
  7: 1,  268,  9436,  143584,  1256080,   7150528,   28254016,   80253952,   166667008,   253854720,    280691712,    219340800,    114831360,     36126720,      5160960;
  8: 1,  526, 26422,  534448,  6080200,  44840992,  231018592,  868063744,  2439304576,  5192901120,   8403141120,  10276761600,   9344563200,   6123479040,   2732728320,    743178240,    92897280;
  9: 1, 1040, 74520, 1974000, 28586280, 265839264, 1727332320, 8240939520, 29766314880, 82936793600, 180085287424, 305735086080, 404516444160, 412765839360, 318459617280, 179601408000, 69827788800, 16721510400, 1857945600;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(8*n-27)*x^2-(12*n-37)*x^3-(8*n-24)*x^4-(2*n-6)*x^5)*P(n-1,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*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[((1 - (2*n - 9)*x - (8*n - 27)*x^2 - (12*n - 37)*x^3 - (8*n - 24)*x^4 - (2*n - 6)*x^5)*P[n - 1, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 1, x], x])/(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] = ((1 - (2*n - 9)*x - (8*n - 27)*x^2 - (12*n - 37)*x^3 - (8*n - 24)*x^4 - (2*n - 6)*x^5)*v[i - 1] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 1], 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

Same fingerprint ?

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