Real-rooted atlas

A178820 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 2\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 24\,x + 24\,x^{2} + 14\,n + 32\,n x + 8\,n x^{2} + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1241640
2200480160840
3112048005760128012960
450403136067200537608960166320
519712161280501760716800430080573441886976
669888709632290304060211206451200309657634406419595520
7230400279552014192640387072006021120051609600206438401966080190356480
87180801013760061501440208158720425779200529858560378470400129761280108134401755198720
921401603446784024330240098402304024979046404087480320423886848025952256007785676805767168015519651840
106150144111288320896163840421724160012792299520259782082563542482944031488737280168689664004498391040299892736132582168576
1117145856344408064311607296016728391680590413824001432737546242424632770562833986355202204211609601049624576002519098982415267266561100484403200
124659200010287513601033224192062321459200250925875200708496588800143273754624020782566604802125489766400146947440640062977474560013740539904076336332808913923665920
1312386304029818880003292004352022042116096099714334720032118512025607557296947200130993147084801662605328384015114593894400940463620096036641439744007328287948803758096384070701890273280
143230924808422686720101384192000746187653120374715973632013561149521920364009802956807341374177280011134417502208012561906925568010277923848192058137751060480207634825216003833258311680182536110080550638859714560
Technical details
Node id
n-8ea454af8ae1e8ed · 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:      24,       16;
  2:     200,      480,       160;
  3:    1120,     4800,      5760,      1280;
  4:    5040,    31360,     67200,     53760,       8960;
  5:   19712,   161280,    501760,    716800,     430080,      57344;
  6:   69888,   709632,   2903040,   6021120,    6451200,    3096576,     344064;
  7:  230400,  2795520,  14192640,  38707200,   60211200,   51609600,   20643840,    1966080;
  8:  718080, 10137600,  61501440, 208158720,  425779200,  529858560,  378470400,  129761280,  10813440;
  9: 2140160, 34467840, 243302400, 984023040, 2497904640, 4087480320, 4238868480, 2595225600, 778567680, 57671680;
%F Formula
The row polynomials satisfy (2*n^2-n+2*n*x)*P(n,x) = (4*n^2+14*n+6+(8*n^2+32*n+24)*x+(8*n+24)*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 + 14*n + 6 + (8*n^2 + 32*n + 24)*x + (8*n + 24)*x^2)*P[n - 1, x])/(2*n^2 - n + 2*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] = ((4*n^2 + 14*n + 6 + (8*n^2 + 32*n + 24)*x + (8*n + 24)*x^2)*v[i - 1])/(2*n^2 - n + 2*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.