Real-rooted atlas

A094638 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121618
26112288406
3247364176614411080
4120521647376149504153600355816
57204089651897625815045593600442368013159376
65040355392584121640213504135961600224280576144506880551164208
740320340761669119424612470784284703360072762163209703636992528482304025796748096
8362880358295048674231689444364288562468416001966772551684042523770884531358269442140353331201334695613760
936288004105497601157642150414995872563210937139072004882608820224136797092741122353947004108822786354446336951268147200075656487286656
10399168005096655360164290114944247275817779221403273331200116706618077184416307726671872973629185589248144233556973977612298915676160004604137832448004663329909134976
11479001600681852211202476050259968425242360135684268414046976002756590815191040119851224026849283560850039662182471527002109378560931077914886144007100439245881344024108939922636800310570249949134848
1262270208009783761817603955265726054476410469036032087382559620864006533895849018163233590385782738534412143043015040040963093067426516697088544903845138923520063307576316232990724369468786558894080135813694897520640022225559260797812736
138717829120014992486871040668207214913536143538890917969921844190895609856001569066025303572480932720029383291699239814301830230310912123219345813966028800274853829898298982400431484106204136865792452947051818396942336285672887398705397760819061052305047552001700993350778908631040
141307674368000244415240294400119121615694540802818320768839024644022251638947020800384075457617514659842595057036695216128001279432606088090877952467012882942706371788812641675801044975616000250963571069707920343043555030356595495980236834053780105489668898816197821159103403550310405265392479103877120000138641415895451371216896
Technical details
Node id
n-2782534ae465360d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d51f33240ea01186

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:       2,        16;
  2:       6,       112,         288;
  3:      24,       736,        4176,         6144;
  4:     120,      5216,       47376,       149504,        153600;
  5:     720,     40896,      518976,      2581504,       5593600,       4423680;
  6:    5040,    355392,     5841216,     40213504,     135961600,     224280576,      144506880;
  7:   40320,   3407616,    69119424,    612470784,    2847033600,    7276216320,     9703636992,     5284823040;
  8:  362880,  35829504,   867423168,   9444364288,   56246841600,  196677255168,   404252377088,   453135826944,   214035333120;
  9: 3628800, 410549760, 11576421504, 149958725632, 1093713907200, 4882608820224, 13679709274112, 23539470041088, 22786354446336, 9512681472000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(16*n^3-16*n^2)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((16*n^3-16*n^2)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(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) = 2 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (16*n^3 - 16*n^2)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((16*n^3 - 16*n^2)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 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] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (16*n^3 - 16*n^2)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((16*n^3 - 16*n^2)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 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.