Real-rooted atlas

A059364 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 16\,n x + 144\,n x^{2} + 4\,n^{2} + 48\,n^{2} x - 288\,n^{2} x^{2} - 4\,n^{3} - 32\,n^{3} x + 144\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(16\,n x^{2} + 64\,n x^{3} - 48\,n^{2} x^{2} - 128\,n^{2} x^{3} + 32\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,n - 16\,n x + 4\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(16\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131619
215176288479
310519526768614414969
494524288131184247808153600557825
510395342768253598473902089433600442368024136635
61351355453904514772642042746883997696003835330561445068801189150527
720270259694368011157549125626544128149335616002187170611216784474112528482304065715834609
834459425190750176025947778464159125307392535197753600105995752243212379703705607913645015042140353331204025540528257
965472907541204690640647515433376468403866214419119555859200478838444728327545935563980873245797646336401232164290569512681472000270717865034467
1013749310575970055155440173062462584961444103538339846942630894272002106847151419392419165577862348854805280860405764550656050855936218115971481600046041378324480019828224058985887
113162341432252473114723632049407912433396846751711788462082589626288291200092445993828753408222584817865568256366944037256560640409861996961071104297362804598374400126714460231434240241089399226368001571113611231870569
127905853580625678995636407200150229420098717601590406980809871369980444104862656004101099536185712640116044253767844290562312830624739413196832538400187362836480317221884533932032002043869319685944115278411561354185932801358136948975206400133905201034440666497
1321345804667687519977299727764400485061923726179920568288637621035315239869255802209376000185463252934225809408603586196789879468032141261789427022482636824006684599654228951042945384038541452902400254780481380270525644814765453863842647900165151980317755152793608190610523050475520012215231574554116332683
146190283353629375627156294560787600165839355537575320802130943142055065978881653460976182477804800859349112298271255347231640420986115271823360851139378990391324835841697386785953994485268482513882775597560627200002734655620518473011036162126173204903090936872961119825869266335679119363583571253551928705024052653924791038771200001187525153219225411332095
Technical details
Node id
n-28f05336e4880951 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a8f71b9105ecef19

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         3,          16;
  2:        15,         176,          288;
  3:       105,        1952,         6768,          6144;
  4:       945,       24288,       131184,        247808,         153600;
  5:     10395,      342768,      2535984,       7390208,        9433600,        4423680;
  6:    135135,     5453904,     51477264,     204274688,      399769600,      383533056,      144506880;
  7:   2027025,    96943680,   1115754912,    5626544128,    14933561600,    21871706112,    16784474112,     5284823040;
  8:  34459425,  1907501760,  25947778464,  159125307392,   535197753600,  1059957522432,  1237970370560,   791364501504,   214035333120;
  9: 654729075, 41204690640, 647515433376, 4684038662144, 19119555859200, 47883844472832, 75459355639808, 73245797646336, 40123216429056, 9512681472000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(32*n^3-48*n^2+16*n)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((32*n^3-48*n^2+16*n)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(16*n^2-16*n)*x)*P(n-1,x) - (16*n^2-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (32*n^3 - 48*n^2 + 16*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((32*n^3 - 48*n^2 + 16*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (16*n^2 - 16*n)*x)*P[n - 1, x] - (16*n^2 - 16*n)*x^2*D[P[n - 1, x], 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] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (32*n^3 - 48*n^2 + 16*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((32*n^3 - 48*n^2 + 16*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (16*n^2 - 16*n)*x)*v[i - 1] - (16*n^2 - 16*n)*x^2*deriv(v[i - 1], x))/(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.