Real-rooted atlas

A290315 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-9 + 4\,n\right)\,P_{n}(x) = \left(-5 + 220\,x + 1980\,x^{2} + 4\,n - 256\,n x - 3184\,n x^{2} + 64\,n^{2} x + 1600\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 160\,x^{2} + 360\,x^{3} + 8\,n x + 208\,n x^{2} - 448\,n x^{3} - 64\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 252\,x - 256\,n x + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-18\,x + 36\,x^{2} + 8\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
31323669
411328524801465
51448861623808840041273
611412632404653447303201814401441757
71432039668461242882402328024781824465696059987357
811306022815006508432050711320012527498889279043201383782402893524529
91392961246536060769024083063672003933875404867601334528381313843204670265600158668300593
1011180206598132852178195201157763960009311083265282989997149440381683482214417095111948801764322560009744944063861
1113542083422987404233109350414463685768001830022875340897529787513984228109502300160226582892358912831294019584007374868300800662515723508917
1211062788175293603633009113020816721287872800315854907570048259160378417830498864741640007681770956143483622414167397101593600436121997341184033790305669120049387067555283817
1313188544890177983225022922513921826754760504804959254346375168594856666458718723437780836094607369906441173312693761410463392953057280933129007276308480245656496126361600168388356584448004005140040620419561
1419565828449584838641858425933958419130757014789607252706465349408012246058119164746241018745906190707507243913474078876612352994460397370678425601157893125522009845766469171659857092608014791875914853273600906706535454720000351025649060430271181
15128697696226255991004135964390601984194013317857117601004409886948162560231967577084814094722675749448739214622721640478592023248247552550013817926747713536010088920623350444553216982473998552560218931247149238627420123750409484044028311896064005245944955130880000033061860745283094943629
Technical details
Node id
n-352c7957093ee52b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dc39590f11d4c375

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,      4;
   3: 1,     32,       36;
   4: 1,    132,      852,        480;
   5: 1,    448,     8616,      23808,         8400;
   6: 1,   1412,    63240,     465344,       730320,       181440;
   7: 1,   4320,   396684,    6124288,     24023280,     24781824,       4656960;
   8: 1,  13060,  2281500,   65084320,    507113200,   1252749888,     927904320,     138378240;
   9: 1,  39296, 12465360,  607690240,   8306367200,  39338754048,   67601334528,   38131384320,    4670265600;
  10: 1, 118020, 65981328, 5217819520, 115776396000, 931108326528, 2989997149440, 3816834822144, 1709511194880, 176432256000;
%F Formula
The row polynomials satisfy (4*n-9)*P(n,x) = (4*n-5+(64*n^2-256*n+220)*x-(256*n^3-1600*n^2+3184*n-1980)*x^2)*P(n-2,x) + ((8*n-10)*x-(64*n^2-208*n+160)*x^2+(128*n^2-448*n+360)*x^3)*d/dx P(n-2,x) - (4-(64*n^2-256*n+252)*x)*P(n-1,x) + ((8*n-18)*x-(16*n-36)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 5 + (64*n^2 - 256*n + 220)*x - (256*n^3 - 1600*n^2 + 3184*n - 1980)*x^2)*P[n - 2, x] + ((8*n - 10)*x - (64*n^2 - 208*n + 160)*x^2 + (128*n^2 - 448*n + 360)*x^3)*D[P[n - 2, x], x] - (4 - (64*n^2 - 256*n + 252)*x)*P[n - 1, x] + ((8*n - 18)*x - (16*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n - 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 5 + (64*n^2 - 256*n + 220)*x - (256*n^3 - 1600*n^2 + 3184*n - 1980)*x^2)*v[i - 2] + ((8*n - 10)*x - (64*n^2 - 208*n + 160)*x^2 + (128*n^2 - 448*n + 360)*x^3)*deriv(v[i - 2], x) - (4 - (64*n^2 - 256*n + 252)*x)*v[i - 1] + ((8*n - 18)*x - (16*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n - 9)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.