Real-rooted atlas

A178820 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 3\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(12 + 36\,x + 27\,x^{2} + 22\,n + 39\,n x + 9\,n x^{2} + 6\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1201232
216030090550
3880288027005407000
43920184803024018900283574375
5152329408022176024192011340013608700000
653760411264127008019958401632960612360612366037500
71766401612800616896012700800149688009797760306180026244048750000
8549120582912026611200678585601047816009879408053887680144342001082565373828125
91633280197683201049241603193344006107270407544275205927644802771366406495390043302602750000000
104685824636979203854822401364014080311351040047636709124903778880330254496013510411202814669001688801419550781250
11130457601968046081337656320539675136014322147840261534873603334569638429422673280173383610406304858560118216098064481508135078125000
12354099205870592004428103680200648448006071345280012889933056019615115520021436519104016550253720086691805200283718635204836113100241805655910888671875
139404416016996761601408942080070849658880240778137600582849146880103119464448013450364928001286191146240882680198400416120664960123804495360193444524008928208806015625000000
14245104640479625216043341742080239520153600903333150720245593700352049542177484807512989552640857460764160072884164953604501669011840192928671936052616910528075889774800325241892039013671875000
Technical details
Node id
n-e34644be6c784048 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:      20,       12;
  2:     160,      300,        90;
  3:     880,     2880,      2700,       540;
  4:    3920,    18480,     30240,     18900,      2835;
  5:   15232,    94080,    221760,    241920,    113400,     13608;
  6:   53760,   411264,   1270080,   1995840,   1632960,    612360,     61236;
  7:  176640,  1612800,   6168960,  12700800,  14968800,   9797760,   3061800,    262440;
  8:  549120,  5829120,  26611200,  67858560, 104781600,  98794080,  53887680,  14434200,  1082565;
  9: 1633280, 19768320, 104924160, 319334400, 610727040, 754427520, 592764480, 277136640, 64953900, 4330260;
%F Formula
The row polynomials satisfy (3*n^2-n+3*n*x)*P(n,x) = (6*n^2+22*n+12+(9*n^2+39*n+36)*x+(9*n+27)*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[((6*n^2 + 22*n + 12 + (9*n^2 + 39*n + 36)*x + (9*n + 27)*x^2)*P[n - 1, x])/(3*n^2 - n + 3*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] = ((6*n^2 + 22*n + 12 + (9*n^2 + 39*n + 36)*x + (9*n + 27)*x^2)*v[i - 1])/(3*n^2 - n + 3*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.