Real-rooted atlas

A178822 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 2\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(10 + 40\,x + 40\,x^{2} + 22\,n + 48\,n x + 8\,n x^{2} + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1362460
242010083361764
331361344016128358436288
41814411289624192019353632256598752
5887047257602257920322560019353602580488491392
63843843902976159667203311616035481600170311681892352107775360
715206401845043293671424255467520397393920340623360136249344129761281256352768

All 15 rows

Technical details
Node id
n-e6c5544535083f78 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:       36,        24;
  2:      420,      1008,        336;
  3:     3136,     13440,      16128,       3584;
  4:    18144,    112896,     241920,     193536,       32256;
  5:    88704,    725760,    2257920,    3225600,     1935360,      258048;
  6:   384384,   3902976,   15966720,   33116160,    35481600,    17031168,     1892352;
  7:  1520640,  18450432,   93671424,  255467520,   397393920,   340623360,   136249344,    12976128;
  8:  5601024,  79073280,  479711232, 1623638016,  3321077760,  4132896768,  2952069120,  1012137984,   84344832;
  9: 19475456, 313657344, 2214051840, 8954609664, 22730932224, 37196070912, 38573703168, 23616552960, 7084965888, 524812288;
%F Formula
The row polynomials satisfy (2*n^2-n+2*n*x)*P(n,x) = (4*n^2+22*n+10+(8*n^2+48*n+40)*x+(8*n+40)*x^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[Together[((4*n^2 + 22*n + 10 + (8*n^2 + 48*n + 40)*x + (8*n + 40)*x^2)*P[n - 1, x])/(2*n^2 - n + 2*n*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^2 + 22*n + 10 + (8*n^2 + 48*n + 40)*x + (8*n + 40)*x^2)*v[i - 1])/(2*n^2 - n + 2*n*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. [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.