Real-rooted atlas

A257612 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(36 - 192\,x + 468\,x^{2} - 440\,x^{3} + 96\,x^{4} - 24\,n + 144\,n x - 488\,n x^{2} + 416\,n x^{3} - 64\,n x^{4} - 32\,n^{2} x + 208\,n^{2} x^{2} - 96\,n^{2} x^{3} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(24\,x - 176\,x^{2} + 360\,x^{3} - 272\,x^{4} + 64\,x^{5} - 16\,n x + 144\,n x^{2} - 224\,n x^{3} + 96\,n x^{4} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-36 + 112\,x - 92\,x^{2} + 16\,x^{3} + 16\,n - 72\,n x + 40\,n x^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 36\,x^{2} - 16\,x^{3} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22832464
31925522088960
412328576732812801618432
5761611523221529684160774432430080
646272139571251959684206080905792465926411796480
727904015807104107497728166222720720650249437568279808128371589120
81677568171462656199359385654234480644377972224114558976096633856167936025613212057600
9100730881808564736342426910721532729077762132376391681022264965121740920422497982668810077184512523197480960
1060455936187303116805570753054723897337888768874740287692871358793318402206750320640257259552768987902259260465152102422830435532800
113627745281916559134728716174544896916487641436163149351366000644098378520535042139049605120004518188830310437343284592649927429529636279500820481088250760396800
122176733184194558102732813256280282726420326155551375361026656912313139220220934780551168167216433812766725931379941703680891961265762304535807217172489956461690882176778240409656254193152819200
13130605875201964736263782419745270313123844311971911619379231006454329630720088653091187208192011021883253386117126168908836123115521554355034693468161716523688996864076289195474944099738779811841306068582481923134162189942784000
147836393472019772744243609628959529705259008884010673160978432882353982660134502435431153032615034880634832478451437895685321329855897298534421107256755601555456390696453867909939232442019513507840010806129526439936998432701972487836414771216384187213954812582297600
Technical details
Node id
n-0c685534a1085bbf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-04cae45d42f04794

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:        4,          2;
  2:       28,         32,           4;
  3:      192,        552,         208,            8;
  4:     1232,       8576,        7328,         1280,           16;
  5:     7616,     115232,      215296,        84160,         7744,           32;
  6:    46272,    1395712,     5195968,      4206080,       905792,        46592,          64;
  7:   279040,   15807104,   107497728,    166222720,     72065024,      9437568,      279808,       128;
  8:  1677568,  171462656,  1993593856,   5423448064,   4377972224,   1145589760,    96633856,   1679360,      256;
  9: 10073088, 1808564736, 34242691072, 153272907776, 213237639168, 102226496512, 17409204224, 979826688, 10077184, 512;
%F Formula
The row polynomials satisfy (2*n-5+4*x)*P(n,x) = -(24*n-36+(32*n^2-144*n+192)*x+(32*n^3-208*n^2+488*n-468)*x^2+(96*n^2-416*n+440)*x^3+(64*n-96)*x^4)*P(n-2,x) - ((16*n-24)*x+(32*n^2-144*n+176)*x^2-(32*n^2-224*n+360)*x^3-(96*n-272)*x^4-64*x^5)*d/dx P(n-2,x) + (16*n-36+(16*n^2-72*n+112)*x+(40*n-92)*x^2+16*x^3)*P(n-1,x) + ((8*n-20)*x-(8*n-36)*x^2-16*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 36 + (32*n^2 - 144*n + 192)*x + (32*n^3 - 208*n^2 + 488*n - 468)*x^2 + (96*n^2 - 416*n + 440)*x^3 + (64*n - 96)*x^4)*P[n - 2, x] - ((16*n - 24)*x + (32*n^2 - 144*n + 176)*x^2 - (32*n^2 - 224*n + 360)*x^3 - (96*n - 272)*x^4 - 64*x^5)*D[P[n - 2, x], x] + (16*n - 36 + (16*n^2 - 72*n + 112)*x + (40*n - 92)*x^2 + 16*x^3)*P[n - 1, x] + ((8*n - 20)*x - (8*n - 36)*x^2 - 16*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 4*x)]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n - 36 + (32*n^2 - 144*n + 192)*x + (32*n^3 - 208*n^2 + 488*n - 468)*x^2 + (96*n^2 - 416*n + 440)*x^3 + (64*n - 96)*x^4)*v[i - 2] - ((16*n - 24)*x + (32*n^2 - 144*n + 176)*x^2 - (32*n^2 - 224*n + 360)*x^3 - (96*n - 272)*x^4 - 64*x^5)*deriv(v[i - 2], x) + (16*n - 36 + (16*n^2 - 72*n + 112)*x + (40*n - 92)*x^2 + 16*x^3)*v[i - 1] + ((8*n - 20)*x - (8*n - 36)*x^2 - 16*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 4*x)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.