Real-rooted atlas

A143491 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 7\,x + 4\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 6 + 17\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
06171235
1249813360315
2120634125310983603465
3720464411956153599846252045045
45040382681208442031611917949640820160675675
5403203515041311164271288433628012497410102895218144011486475
636288035667361531551637527596573940495610470034234668119224801814400218243025
73628800396590401923892565437466369867440461192381539959498380495806148149291280199584004583103525

All 15 rows

Technical details
Node id
n-3b84ef3ac5bb741d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2d1227ef2afb39a7

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         6,         17,          12;
  1:        24,         98,         133,           60;
  2:       120,        634,        1253,         1098,          360;
  3:       720,       4644,       11956,        15359,         9846,         2520;
  4:      5040,      38268,      120844,       203161,       191794,        96408,        20160;
  5:     40320,     351504,     1311164,      2712884,      3362801,      2497410,      1028952,       181440;
  6:    362880,    3566736,    15315516,     37527596,     57394049,     56104700,     34234668,     11922480,      1814400;
  7:   3628800,   39659040,   192389256,    543746636,    986744046,   1192381539,    959498380,    495806148,    149291280,     19958400;
  8:  39916800,  479795040,  2592190296,   8289884068,  17379144138,  24957125481,  24863060648,  16967848188,   7591877856,   2011037760,   239500800;
  9: 479001600, 6276458880, 37343619072, 133177082664, 316318222540, 525414379566, 622799359029, 526833966680, 311684560716, 122826865248, 29017500480, 3113510400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+3+(2*n+7)*x+(n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 17*x + 12*x^2.
%t Mathematica
P[0, x_] := 6 + 17*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3 + (2*n + 7)*x + (n + 4)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 6 + 17*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 3 + (2*n + 7)*x + (n + 4)*x^2)*v[i - 1])/(1 + x)); 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. [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. [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.