Real-rooted atlas

A156992 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 6\,n + 16\,n x + 8\,n x^{2} + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112820
212028896504
3134457606912153615552
41728010752023040018432030720570240
5253440207360064512009216000552960073728024261120
6419328042577920174182400361267200387072000185794560206438401175731200
7774144009392947204768727040130056192002023096320017340825600693633024066060288063959777280

All 15 rows

Technical details
Node id
n-9754b486dd6d7f50 · 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:          12,            8;
  2:         120,          288,            96;
  3:        1344,         5760,          6912,           1536;
  4:       17280,       107520,        230400,         184320,          30720;
  5:      253440,      2073600,       6451200,        9216000,        5529600,         737280;
  6:     4193280,     42577920,     174182400,      361267200,      387072000,      185794560,       20643840;
  7:    77414400,    939294720,    4768727040,    13005619200,    20230963200,    17340825600,     6936330240,      660602880;
  8:  1579253760,  22295347200,  135258439680,   457797795840,   936404582400,  1165303480320,   832359628800,   285380444160,    23781703680;
  9: 35300966400, 568531353600, 4013162496000, 16231012761600, 41201801625600, 67421129932800, 69918208819200, 42807066624000, 12842119987200, 951268147200;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (4*n^2+6*n+2+(8*n^2+16*n+8)*x+(8*n+8)*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 + 6*n + 2 + (8*n^2 + 16*n + 8)*x + (8*n + 8)*x^2)*P[n - 1, x])/(2*n - 1 + 2*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 + 6*n + 2 + (8*n^2 + 16*n + 8)*x + (8*n + 8)*x^2)*v[i - 1])/(2*n - 1 + 2*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.