Real-rooted atlas

A196347 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21612230
3120144546324
4115219201152288244536
51344028800240009600180012077760
618432048384051840028800086400129607201574640
7290304090316801185408084672003528000846720105840504036741600
85160960018579456028901376025288704013547520045158400903168096768040320969978240
9102187008041803776007524679680780337152051209625602194698240609638400104509440979776036288028570268160
102229534720010218700800020901888000025082265600019508428800010241925120036578304000870912000013063680001088640003628800928533715200
115313724416002697737011200618231398400084304281600007587385344000472103976960020654548992006322821120001317254400001756339200013172544003991680032998659724800
12137339338752007651763159040019423706480640029675107123200030349541376000021851669790720011330495447040042489357926400113810780160002107607040000252912844800172440576004790016001272805446528000
133825881579520002321034824908800646573986938880010942021317427200125377327595520001025814498508800061548869910528002735505329356800897587686195200213711353856000356185589760003885660979200242853811200622702080052948706575564800
14114266329841664007498727895859200022746141284106240042242833813340160053615904455393280049147912417443840033509940284620800017233683574947840067019880569241600195474651660288004188742535577600634657959936000634657959936003661488230400871782912002362836030934579200
Technical details
Node id
n-14414e2c5a7f4554 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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:          3,          1;
  2:         16,         12,          2;
  3:        120,        144,         54,          6;
  4:       1152,       1920,       1152,        288,         24;
  5:      13440,      28800,      24000,       9600,       1800,        120;
  6:     184320,     483840,     518400,     288000,      86400,      12960,       720;
  7:    2903040,    9031680,   11854080,    8467200,    3528000,     846720,    105840,      5040;
  8:   51609600,  185794560,  289013760,  252887040,  135475200,   45158400,   9031680,    967680,   40320;
  9: 1021870080, 4180377600, 7524679680, 7803371520, 5120962560, 2194698240, 609638400, 104509440, 9797760, 362880;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = (2*n^2+4*n+(n^2+4*n)*x+n*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[((2*n^2 + 4*n + (n^2 + 4*n)*x + n*x^2)*P[n - 1, x])/(n + 1 + 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] = ((2*n^2 + 4*n + (n^2 + 4*n)*x + n*x^2)*v[i - 1])/(n + 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. [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.