Real-rooted atlas

A049374 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(6 + x + n\right)\,P_{n-1}(x) + \left(5 + 5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
26120182
36493513911040
482456304114664115760
5122215120569313302830951277040
620694252485188852279109640589513215522560
73932846555325599237322054040099307475109411751122744960
88272265051327836896685500004147044576147677747394241867622413003134080
9190475820553423896446520671005636539219276468240226644905014158768429916279180078670080
1047595613520594477589218065256800246120147702664031081299690257113836011898553831245604558534012308551650560
11128151337593852778670578777521586267358227772145884874114157635741701426386769508272818246283286058573820566715407171476529914880
123696059366078058677720404604807480240745492463047096808500806508644159095177974112304005496489647882351650752061910412399616009444648012363569379921920
1311361372997850695286763756531517852713111541513981412409882055270986303801618365716942466200467219135553288268212184430453578860491922777126136882316504774130026559183078883672473600
143706093382227728259995819184780434601028758068936410995522237311022767704144547079518362521232144050419521242266877076835371139409714647632553058149944314330613497224238544332629026417481164413091360084349747200
Technical details
Node id
n-eddbec3320e244e5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-24aa8c9a1e22e667

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:           7,           1;
  2:          61,          20,           1;
  3:         649,         351,          39,          1;
  4:        8245,        6304,        1146,         64,         1;
  5:      122215,      120569,       31330,       2830,        95,        1;
  6:     2069425,     2485188,      852279,     109640,      5895,      132,       1;
  7:    39328465,    55325599,    23732205,    4040099,    307475,    10941,     175,     1;
  8:   827226505,  1327836896,   685500004,  147044576,  14767774,   739424,   18676,   224,   1;
  9: 19047582055, 34238964465, 20671005636, 5392192764, 682402266, 44905014, 1587684, 29916, 279, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+6+x)*P(n-1,x) + (5+5*x)*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[(n + 6 + x)*P[n - 1, x] + (5 + 5*x)*D[P[n - 1, x], x]];
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] = (n + 6 + x)*v[i - 1] + (5 + 5*x)*deriv(v[i - 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

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.