Real-rooted atlas

A094638 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} - n^{3} - 4\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
2141630
392232144468
46522632350415368324
5511228832605045593619200169584
64442432451294250413596169734402764803920976
74259523839968143547842847033631580800185656324515840101753312
8447868848190176221352288562468416853633920773442048387201024825753602933341920
9513187206431345283514657632109371390722119187856026172913152201142932488791031808167215104093088517760
10637081920912722860857955269792214032733312506539140960796507130112831958532608556456624128216191877120371589120003226564530560
118523152640137558347776996661781568426841404697611964369857600229307188826883042718539750427595294023680163666039726085730597273600899245670400121325172407040
121222970227202197369847808179087036803208738255962086428358922955808064267319737382410376135388828161193313050353664957838790283264510940537159680162977961738240235438866432004920101122164480
13187406085888037122623050752336419275588992184419089560985668101824014912001784540872544563234021009864698752475383278603264004831414978681036834824065906196480168945908271022084952262055034880663152807116800214082757089331200
143055190503680066178675385856066054393019664644022251638947020816669941736871296049650325957178880010932651663350385921801747233575256064222216957440243712020254678708492697601326000092317384704590336886998827008160165907776143360199966077222912009949872311267481600
Technical details
Node id
n-a63ba5af2d4957e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-23394a58ed9a04a5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        2;
  2:       14,        16;
  3:       92,       232,        144;
  4:      652,      2632,       3504,        1536;
  5:     5112,     28832,      60504,       55936,       19200;
  6:    44424,    324512,     942504,     1359616,      973440,      276480;
  7:   425952,   3839968,   14354784,    28470336,    31580800,    18565632,     4515840;
  8:  4478688,  48190176,  221352288,   562468416,   853633920,   773442048,   387201024,   82575360;
  9: 51318720, 643134528, 3514657632, 10937139072, 21191878560, 26172913152, 20114293248, 8791031808, 1672151040;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (4*n^2 - 4*n)*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (4*n^2 - 4*n)*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.