Real-rooted atlas

A094638 → D → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-16\,n x^{2} + 8\,n^{2} x + 32\,n^{2} x^{2} - n^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
271623
346232288566
432626327008614416110
5255628832121008223744153600529740
622212324512188500854384647787520442368019881396
7212976383996828709568113881344252646400297050112144506880840847248
8223934448190176442704576224987366468290713601237507276812390432768528482304039622407696
9256593606431345287029315264437485562881695350284804187666104326436573839365626260357122140353331202060067057120
1031854096091272286081159105395848561309332484052313127680127441140817922662267304345635613223944192276725602713609512681472000117199053182880
114261576320137558347776199332356313617073656187904957149588608003668915021230089736699327201281766098817515520209492530849382414670329020416004604137832448007243956004674816
126114851136021973698478083581740736064034953023848345622687138364646401028277115798118433203633244250112763720352226344961226033651562577921308007775128780808344471640997888024108939922636800483472558627285248
1393703042944037122623050752672838551177984737676358243942454481459211929600285526539607130112108867231567036006430424529830608896006184211172711727104891496087198629888086500305034763304965071116344355717120135813694897520640034657676800842686976
1415275952518400661786753858560132108786039329281608900655578808321333595338949703680794405215314862080034984485322721234944115311822948816388096284437705523511951360518519774937413058560678912047266500968448604504972286798856192328019779125541601280819061052305047552002656049118140775329280
Technical details
Node id
n-c82f794c6974cbe9 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-6dec8cea3983a758

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:        1;
  2:        7,        16;
  3:       46,       232,        288;
  4:      326,      2632,       7008,        6144;
  5:     2556,     28832,     121008,      223744,       153600;
  6:    22212,    324512,    1885008,     5438464,      7787520,      4423680;
  7:   212976,   3839968,   28709568,   113881344,    252646400,    297050112,    144506880;
  8:  2239344,  48190176,  442704576,  2249873664,   6829071360,  12375072768,  12390432768,   5284823040;
  9: 25659360, 643134528, 7029315264, 43748556288, 169535028480, 418766610432, 643657383936, 562626035712, 214035333120;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*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.