Real-rooted atlas

A088729 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(7 + 3\,x\right)\,P_{n-1}'(x) + \left(2 + 2\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 23 + 11\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
12311135
2173118211313
3160213893613413387
41757518231614585550142857
5222497266062109096199001730691619355
631888064287459205524246051652815314391110048073
750988405757215674130446710914050154674112189852781161180602523
88992224571455488380886051671268493569451209364410903253554834614413559049961

All 15 rows

Technical details
Node id
n-c8f23df3c6efe849 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-43d131bfc3713e5e

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 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:           23,           11,            1;
   2:          173,          118,           21,            1;
   3:         1602,         1389,          361,           34,           1;
   4:        17575,        18231,         6145,          855,          50,          1;
   5:       222497,       266062,       109096,        19900,        1730,         69,         1;
   6:      3188806,      4287459,      2055242,       460516,       52815,       3143,        91,        1;
   7:     50988405,     75721567,     41304467,     10914050,     1546741,     121898,      5278,      116,      1;
   8:    899222457,   1455488380,    886051671,    268493569,    45120936,    4410903,    253554,     8346,    144,     1;
   9:  17329515172,  30258494023,  20259195331,   6899192860,  1336459510,  155491665,  11115237,   486540,  12585,   175,   1;
  10: 362164300173, 676594803611, 492598839953, 185664263631, 40611334115, 5465424679, 466146219, 25418217, 875325, 18260, 209, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (7+3*x)*d/dx P(n-1,x) + (2+2*x)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 23 + 11*x + x^2.
%t Mathematica
P[1, x_] := 23 + 11*x + x^2;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (7 + 3*x)*D[P[n - 1, x], x] + (2 + 2*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] = 23 + 11*x + x^2; for(n = 2, N, my(i = n); v[i] = (4 + x)*v[i - 1] + (7 + 3*x)*deriv(v[i - 1], x) + (2 + 2*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

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.