Real-rooted atlas

A039756 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-8 - 8\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2138122
37973191172
46017603224811732
55339861751341338131120560
653861105840813793040854713761277336
76075271409089132961764856317046122419110714188784
875606252024808022696900137109764769846938320932203296170021264
9102637235312078385405898876293245364127597474337624545171804396820985911280798272
1015062250855130155384760311795764040444323354873890112156316823509393028825712173755329544125385666656
1123726435583895362779131490600707671435738849978810175255035693541434954748532616299077221631642037830909885951541040440000
1239885224909716526617247443055350790022331508461790423345406758391113101782640363193720884799106600961133155259993928909636230822265744112325003559488
13712017090599532150147751065653864664901867896434964702262818809275517344980124958711331111093687635967457046766640782623577936043661555009443621713214867971871298594924969216
14134408217821205657118094918944145868149536535119415816715154161723163456089513107209423870284047822622952608315356890955761635067038363295551565791451256164402971732776471544823922659239151217660020256590208
Technical details
Node id
n-cfca73442782a1b8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-edeb94a827b82f42

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,         1;
  2:        13,         8,         1;
  3:        79,        73,        19,         1;
  4:       601,       760,       322,        48,         1;
  5:      5339,      8617,      5134,      1338,       131,        1;
  6:     53861,    105840,     81379,     30408,      5471,      376,       1;
  7:    607527,   1409089,   1329617,    648563,    170461,    22419,    1107,      1;
  8:   7560625,  20248080,  22696900,  13710976,   4769846,   938320,   93220,   3296,    1;
  9: 102637235, 312078385, 405898876, 293245364, 127597474, 33762454, 5171804, 396820, 9859, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-1+(2*n-1)*x)*P(n-1,x) - (8+8*x+2*x^2)*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[(4*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (8 + 8*x + 2*x^2)*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] = (4*n - 1 + (2*n - 1)*x)*v[i - 1] - (8 + 8*x + 2*x^2)*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.