Real-rooted atlas

A105278 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(4 + 2\,x\right)\,P_{n-1}'(x) + \left(1 + x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3138122
47363151152
55015441862411256
640515225229043035112032
73763355656294157160855481130768
83943536530234007011187551851515336311586944
9459655383790085817700201857638563041888254880121241984
1058941091116780049900648363562028480124661071126857643996991310579712
1182407314117572114001484548605656418960169175370267361922642850162480598512014920975104
121247016223328394129951259946973551266395344536660812106662839267808770259403302892458635143183939634176
1320297640121349037150620848230114796625594623076081984834855167940020162269620276205148592123814354892801207816811532861774848
14353501752440390135227964739457329713910541727264106618976281030854314825968316589206795668819407804943251172424493579307816458195129825546764288
156557380318692117567956449226419549727557005111997602230650445527247686921113541150442721929281677323228131306832189639480031108379272450460011239784219312241615785559906304
Technical details
Node id
n-54d11b5bf400eddf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-747352d83647270b

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        3,         1;
   3:       13,         8,        1;
   4:       73,        63,       15,        1;
   5:      501,       544,      186,       24,       1;
   6:     4051,      5225,     2290,      430,      35,       1;
   7:    37633,     55656,    29415,     7160,     855,      48,     1;
   8:   394353,    653023,   400701,   118755,   18515,    1533,    63,    1;
   9:  4596553,   8379008,  5817700,  2018576,  385630,   41888,  2548,   80,  1;
  10: 58941091, 116780049, 90064836, 35620284, 8012466, 1071126, 85764, 3996, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (4 + 2*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (4 + 2*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.