Real-rooted atlas

A008955 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 4\,n^{2} x - n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x + 4\,n^{3} x^{2} - 8\,n^{4} - 14\,n^{4} x - 6\,n^{4} x^{2} + 4\,n^{5} + 8\,n^{5} x + 4\,n^{5} x^{2} - n^{6} - 2\,n^{6} x - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 2\,n^{2} x - 4\,n^{3} - 4\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
213821
3220314108642
45340123789372230429394
5181340588828707847372304720001922319
68300780348431165798705147751904194176803110400171410931
74951736002555632240546032157661760825323895688120129727958417781120020057988852
837419564000229498316400600261597480867392805732747306330360383512020576108421270272130056192002986817524020
93498923298000248808313128007710549643596013595446259786414910264066988510408616099822445132430106736111031726510081185137049600552049255120077
103969746630980003212748622192800115197798369519602401213358051226432053674532391385284050191412068241670176089057313662794124499502081368805765248000131681894400000124081991376524577
115375723743405600048802320037723760019885270150361323204788004777184079088754238349700422890081196058773304616306047456697416952580307570325347603904010217793231097280642000935217779200001752686014464000033342861260291475222
128568129508286920000862454484357075800003937494284617225080001076057879578628873200195548646695874742634024806667204952476902702240975008516975677940144121555967178699968064642192891402889376019249825235535489024034236208570595328000275331039363072000010558874341860713507430
13158818238876257740000017557754007871267560000888006240169446978900002716547107417417135800005597499264439724119413008182859979834260854076528700705398388655348480356778349610749110923134403839028361979834152533601541083762187113125024004160558056986167985561667799746742960701440005040852445672243200003892443549401915083015803
1433881822885148602780000040814450359455700501200002265570260965225814697000076722541092551331957180000176826139641189284979536100292855524302357250113901204358518627887133275228279319328429288255663364991707520225091467648020628660025440113964856664869183440371200414220652850169099070737921023168460177402318385971215389194156886087516160001064007623917279641600001652783481431104903402600287
Technical details
Node id
n-95748ec646761472 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a2a8c4df3dfa7066

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:            13,              8;
  3:           220,            314,            108;
  4:          5340,          12378,           9372,            2304;
  5:        181340,         588828,         707847,          372304,           72000;
  6:       8300780,       34843116,       57987051,        47751904,        19417680,         3110400;
  7:     495173600,     2555632240,     5460321576,      6176082532,      3895688120,      1297279584,      177811200;
  8:   37419564000,   229498316400,   600261597480,    867392805732,    747306330360,    383512020576,   108421270272,    13005619200;
  9: 3498923298000, 24880831312800, 77105496435960, 135954462597864, 149102640669885, 104086160998224, 45132430106736, 11103172651008, 1185137049600;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(2*n^6-8*n^5+14*n^4-12*n^3+4*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*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^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*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^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*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

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.