Real-rooted atlas

A225476 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
22417
361813138
42496116401277
5120600102066012112522
67204320948091203542364127547
750403528094920121800744661843810931351038
84032032256010281601653120139406457657694376328015112457
936288032659201203552023284800251546401485288043191604784409841183764082
103628800362880001518048003435264004536302403509654401509525603166944024118822952411524907087
11399168004390848002055715200532889280083283076808002985760461538264014852508002289217261211845888573130536665238
124790016005748019200297779328008701862400015721763904018094551552013173111072058195463040142829441041638807456607694362657201667096092637
136227020800809512704004597617024001495522828800306831260736041202639878403636300507840206167721760071156481396013510172676011653960500304378620797161115787642820042
1487178291200122049607680075409221888002702527027200062089105478400953803154304009913489185600068976248140800310711927125608502721587360126253138284082490859360152348742223914841402374890155427
Technical details
Node id
n-874228f7413096c5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-da8d2afb3d35ceef

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:      1,       1;
  2:      2,       4,        1;
  3:      6,      18,       13,        1;
  4:     24,      96,      116,       40,        1;
  5:    120,     600,     1020,      660,      121,        1;
  6:    720,    4320,     9480,     9120,     3542,      364,       1;
  7:   5040,   35280,    94920,   121800,    74466,    18438,    1093,      1;
  8:  40320,  322560,  1028160,  1653120,  1394064,   576576,   94376,   3280,    1;
  9: 362880, 3265920, 12035520, 23284800, 25154640, 14852880, 4319160, 478440, 9841, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+(2*n-1)*x)*P(n-1,x) - (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[(n + (2*n - 1)*x)*P[n - 1, x] - (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] = (n + (2*n - 1)*x)*v[i - 1] - (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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.