Real-rooted atlas

A156992 → basis A008277 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(3 + x + 3\,n + n x\right)\,P_{n-1}(x) + \left(1 + x + n + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
260426108
3888960288242160
4181202652012360216012059280
5485280893520568800154800180007202121120
6164455203637872029413440109872001990800166320504095387040
768572224017662982401731260160826398720206035200268128001693440403205244261120
8344111846401009209196801161161568006762413952021624382080387628416038102400018869760362880344973323520
920415447360006703271769600884374122240060835852224002383762348800549207993600743758848005726246400228614400362880026685447667200
1014110696575360051196325685120076090006903680060368332147200028067557795200079460497670400138454214976001468139904000910103040002993760000399168002393197294118400
11112234098493440004452912157478400073530162797414400660684277161216003558857826585600011993185059609600257215524572160035105883563520029954365056000152801510400042152140800479001600245887624355788800
1210165915645966080004373202882663014400793500784323287040079598355480483840004877007211377100800191057087282181120048879992356715520082190920792780800902601683343360063325688025600027056405376000635156121600622702080028652893738601932800
131039216860703397376004813118629440016896009509884824016152576001052828699678442086400723124365485160038400323321232294963302400963914928266594304001936084172627404800026204698503599616002365392883037184001388767614474240050432641459200010199860070400871782912003754120074855716966400
1411896153817325313536000589671915397436551680001259211056098110635520001524068721361936189440001159695812932406830080005832064771207234560000019895558397282247680000466445503338419404800075472295512450636800083967271071780096000633940003118822400031605966404812800098768645015040001739206909440001307674368000548886921277037511168000
Technical details
Node id
n-496eebcad55bac71 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f80b58a897df2b0a

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:             6,             2;
  2:            60,            42,             6;
  3:           888,           960,           288,            24;
  4:         18120,         26520,         12360,          2160,           120;
  5:        485280,        893520,        568800,        154800,         18000,          720;
  6:      16445520,      36378720,      29413440,      10987200,       1990800,       166320,        5040;
  7:     685722240,    1766298240,    1731260160,     826398720,     206035200,     26812800,     1693440,      40320;
  8:   34411184640,  100920919680,  116116156800,   67624139520,   21624382080,   3876284160,   381024000,   18869760,    362880;
  9: 2041544736000, 6703271769600, 8843741222400, 6083585222400, 2383762348800, 549207993600, 74375884800, 5726246400, 228614400, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3+(n+1)*x)*P(n-1,x) + (n+1+(n+1)*x)*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[(3*n + 3 + (n + 1)*x)*P[n - 1, x] + (n + 1 + (n + 1)*x)*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] = (3*n + 3 + (n + 1)*x)*v[i - 1] + (n + 1 + (n + 1)*x)*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. [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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.