Real-rooted atlas

A081578 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(108\,n x + 324\,n x^{2} + 162\,n x^{3} + 162\,n^{2} x^{2} + 243\,n^{2} x^{3} + 81\,n^{3} x^{3}\right)\,P_{n}(x) = \left(192 + 1404\,x + 3348\,x^{2} + 2916\,x^{3} + 972\,x^{4} - 192\,n - 756\,n x + 54\,n x^{2} + 2349\,n x^{3} + 1458\,n x^{4} - 432\,n^{2} x - 1566\,n^{2} x^{2} - 810\,n^{2} x^{3} + 486\,n^{2} x^{4} - 216\,n^{3} x^{2} - 567\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 672\,x + 2628\,x^{2} + 4590\,x^{3} + 3240\,x^{4} + 486\,x^{5} + 192\,n x + 1584\,n x^{2} + 4428\,n x^{3} + 4455\,n x^{4} + 729\,n x^{5} + 144\,n^{2} x^{2} + 864\,n^{2} x^{3} + 1377\,n^{2} x^{4} + 243\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
27422776
320180324108632
461732210618364055140
5182273010908163088910145840496
6547984650301107028100845393665103313036
7164034440214488595080813240542376163296174962382056
8492111810486778029728085364630521964026535606473525904917907844
914762398574337543213775832310116604016206829760696121247282480058196830133300640
104428713286101274219160385176163220670265563036262855530155521944526126599251010649539984174652
111328604384380469701722535717968008988401574132616196616548815547470487603149242192686203377602821257647216488536
1239858114348916169832862102908610037216871558590973400130187736001302085562485003987953526799940884672118121168548690873352605904372
13119574246633938604335492406186110016557826410439521258787853284303295565164040789689439544364921763015679871460347562414042834144622320522381546304784
1435872271506635342121732657156663168127108991203521347956830644015966813763335574237663780767612144611263621021297149685967316568588133545808891495474974717445352755157369260
Technical details
Node id
n-f3cdfd035a79c74f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2d7feb89f1e92983

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,      6;
  2:     7,     42,      27;
  3:    20,    180,     324,      108;
  4:    61,    732,    2106,     1836,      405;
  5:   182,   2730,   10908,    16308,     8910,     1458;
  6:   547,   9846,   50301,   107028,   100845,    39366,     5103;
  7:  1640,  34440,  214488,   595080,   813240,   542376,   163296,    17496;
  8:  4921, 118104,  867780,  2972808,  5364630,  5219640,  2653560,   647352,   59049;
  9: 14762, 398574, 3375432, 13775832, 31011660, 40162068, 29760696, 12124728, 2480058, 196830;
%F Formula
The row polynomials satisfy (108*n*x+(162*n^2+324*n)*x^2+(81*n^3+243*n^2+162*n)*x^3)*P(n,x) = -(192*n-192+(432*n^2+756*n-1404)*x+(216*n^3+1566*n^2-54*n-3348)*x^2+(567*n^3+810*n^2-2349*n-2916)*x^3-(486*n^2+1458*n+972)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2628)*x^2+(864*n^2+4428*n+4590)*x^3+(1377*n^2+4455*n+3240)*x^4+(243*n^2+729*n+486)*x^5)*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 - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*D[P[n - 1, x], x])/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)]];
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 - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*deriv(v[i - 1], x))/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)); 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.