Real-rooted atlas

A266213 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-96 - 648\,x - 1242\,x^{2} - 324\,x^{3} + 486\,x^{4} + 72\,n x - 108\,n x^{2} - 810\,n x^{3} - 729\,n x^{4} + 216\,n^{2} x^{2} + 162\,n^{2} x^{3} + 243\,n^{2} x^{4} + 162\,n^{3} x^{3}\right)\,P_{n}(x) = \left(96 + 648\,x + 1242\,x^{2} + 324\,x^{3} - 486\,x^{4} - 192\,n - 792\,n x - 648\,n x^{2} + 486\,n x^{3} + 729\,n x^{4} - 432\,n^{2} x - 1512\,n^{2} x^{2} - 1134\,n^{2} x^{3} - 243\,n^{2} x^{4} - 216\,n^{3} x^{2} - 486\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 672\,x + 2664\,x^{2} + 4860\,x^{3} + 3888\,x^{4} + 972\,x^{5} + 192\,n x + 1584\,n x^{2} + 4752\,n x^{3} + 6318\,n x^{4} + 3645\,n x^{5} + 729\,n x^{6} + 144\,n^{2} x^{2} + 864\,n^{2} x^{3} + 1458\,n^{2} x^{4} + 486\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
231215
374854109
417168378216779
541528178221608105321
699157270741317610530291635367
723945122538063936793804665610206230309
85771262485212270864458460419904193914349921476547
913933465627291610480322251800282268820309947698241180989350401
103363937568438043799440990792015822216156049749202896295245039366058624479
1181192507042538378131137924024566078242112984981067964179239680928110224801299078364541149
12196016640567470630435557521538238603528593285418263345596970403821651281645498804027141842515282251154555
1347321174513621596058140261328560404170148261104026910058143394433952296177974217462757606612307021445519521381746613819760441
141142434555860615041104403451601963946250588930231612347984250184325608801965091671014803584300766845743425891805525112462424464104484408339351
Technical details
Node id
n-befb1852857a5725 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2947906bdaa38e4f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  2:    3,    12;
  3:    7,    48,     54;
  4:   17,   168,    378,     216;
  5:   41,   528,   1782,    2160,     810;
  6:   99,  1572,   7074,   13176,   10530,    2916;
  7:  239,  4512,  25380,   63936,   79380,   46656,   10206;
  8:  577, 12624,  85212,  270864,  458460,  419904,  193914,  34992;
  9: 1393, 34656, 272916, 1048032, 2251800, 2822688, 2030994, 769824, 118098;
%F Formula
The row polynomials satisfy -(96-(72*n-648)*x-(216*n^2-108*n-1242)*x^2-(162*n^3+162*n^2-810*n-324)*x^3-(243*n^2-729*n+486)*x^4)*P(n,x) = -(192*n-96+(432*n^2+792*n-648)*x+(216*n^3+1512*n^2+648*n-1242)*x^2+(486*n^3+1134*n^2-486*n-324)*x^3+(243*n^2-729*n+486)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2664)*x^2+(864*n^2+4752*n+4860)*x^3+(1458*n^2+6318*n+3888)*x^4+(486*n^2+3645*n+972)*x^5+729*n*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*D[P[n - 1, x], x])/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*deriv(v[i - 1], x))/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)); 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. [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.