Real-rooted atlas

A151775 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(24 + 32\,x + 10\,x^{2}\right)\,P_{n-1}'(x) + \left(16 + 32\,x + 20\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2577624157
327635462348072012425
4250737655576627984262080403201836697
53658152311894052215199176095588640296352003628800436366445
67828053417304356678764861576394440879776960191055110404710182400479001600152053957237
7230964463548310450328632382200276339936402108904773976013187071497600490014645120010025503488008717829120073053601590065

All 15 rows

Technical details
Node id
n-460e179a989db7f3 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2285acaa57b31e0b

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:                  3,                   2;
  2:                 57,                  76,                  24;
  3:               2763,                5462,                3480,                 720;
  4:             250737,              655576,              627984,              262080,               40320;
  5:           36581523,           118940522,           151991760,            95588640,            29635200,             3628800;
  6:         7828053417,         30435667876,         48615763944,         40879776960,         19105511040,          4710182400,           479001600;
  7:      2309644635483,      10450328632382,      20027633993640,      21089047739760,      13187071497600,       4900146451200,       1002550348800,        87178291200;
  8:    898621108880097,    4638024381708976,   10367787111743904,   13117171658048640,   10278725669372160,    5111018283571200,    1575660435148800,    275483400192000,    20922789888000;
  9: 445777636063460643, 2584568731265561042, 6601458159781401120, 9753008378506390080, 9188659740573331200, 5727242569227379200, 2362582180204646400, 622221802339737600, 94968543301632000, 6402373705728000;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + (24+32*x+10*x^2)*d/dx P(n-1,x) + (16+32*x+20*x^2+4*x^3)*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[(3 + 2*x)*P[n - 1, x] + (24 + 32*x + 10*x^2)*D[P[n - 1, x], x] + (16 + 32*x + 20*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];
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] = (3 + 2*x)*v[i - 1] + (24 + 32*x + 10*x^2)*deriv(v[i - 1], x) + (16 + 32*x + 20*x^2 + 4*x^3)*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 - 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.