Real-rooted atlas

A038320 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(22 + 12\,x + 11\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 132 + 72\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
013272204
1217823766485202
23194452272285125184117912
3439230958320784080285120388802505630
4579783615812280172497609408960256608027993651114852
574405562243509112332057880241496640987940802155507219595521013777898
693538420835714669765844218688531292608028979596809484231681724405761343692819696256304
711575379574505107472329642960835210519593638471724502080312979645448535808512133025587290699264376690901814

All 15 rows

Technical details
Node id
n-12372a4647df96de · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-57ca34e86b523c55

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           132,            72;
  1:          2178,          2376,            648;
  2:         31944,         52272,          28512,           5184;
  3:        439230,        958320,         784080,         285120,          38880;
  4:       5797836,      15812280,       17249760,        9408960,        2566080,         279936;
  5:      74405562,     243509112,      332057880,      241496640,       98794080,       21555072,       1959552;
  6:     935384208,    3571466976,     5844218688,     5312926080,     2897959680,      948423168,     172440576,      13436928;
  7:   11575379574,   50510747232,    96429608352,   105195936384,    71724502080,    31297964544,    8535808512,    1330255872,     90699264;
  8:  141476861460,  694522774440,  1515322416960,  1928592167040,  1577939045760,   860694024960,  312979645440,   73164072960,   9976919040,   604661760;
  9: 1711870023666, 9337472856360, 22919251556520, 33337093173120, 31821770756160, 20828795404032, 9467634274560, 2950950942720, 603603601920, 73164072960, 3990767616;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 132 + 72*x.
%t Mathematica
P[0, x_] := 132 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (6*n + 12)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 132 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (6*n + 12)*x)*v[i - 1])/(n + 1)); 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.