Real-rooted atlas

A059473 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(3 - 3\,x - 7\,x^{2} + 4\,x^{4} + 18\,n x - 3\,n x^{2} - 27\,n x^{3} - 8\,n x^{4} + 33\,n^{2} x^{2} + 15\,n^{2} x^{3} - 17\,n^{2} x^{4} - 4\,n^{2} x^{5} + 18\,n^{3} x^{3} + 6\,n^{3} x^{4} - 4\,n^{3} x^{5} + 3\,n^{4} x^{4}\right)\,P_{n}(x) = \left(6 + 42\,x + 106\,x^{2} + 104\,x^{3} - 64\,x^{5} - 32\,x^{6} + 36\,n x + 198\,n x^{2} + 366\,n x^{3} + 204\,n x^{4} - 88\,n x^{5} - 80\,n x^{6} + 66\,n^{2} x^{2} + 270\,n^{2} x^{3} + 314\,n^{2} x^{4} + 44\,n^{2} x^{5} - 64\,n^{2} x^{6} + 36\,n^{3} x^{3} + 108\,n^{3} x^{4} + 64\,n^{3} x^{5} - 16\,n^{3} x^{6} + 6\,n^{4} x^{4} + 12\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(12 + 36\,x + 8\,x^{2} - 100\,x^{3} - 132\,x^{4} - 64\,x^{5} - 16\,x^{6} + 72\,n x + 192\,n x^{2} + 120\,n x^{3} - 112\,n x^{4} - 144\,n x^{5} - 48\,n x^{6} + 132\,n^{2} x^{2} + 300\,n^{2} x^{3} + 160\,n^{2} x^{4} - 60\,n^{2} x^{5} - 48\,n^{2} x^{6} + 72\,n^{3} x^{3} + 120\,n^{3} x^{4} + 32\,n^{3} x^{5} - 16\,n^{3} x^{6} + 12\,n^{4} x^{4} + 12\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
218144144306
380960201610244080
435656961828819456640050196
515843168013708822425615360036864585072
670481691529213122040832215040010690562007046558504
73136087808057669121613824023065600176209926823936104857671373696

All 15 rows

Technical details
Node id
n-eeaf7e74dfbcc8d5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8d5c66df15415afb

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,       16;
  2:     18,      144,       144;
  3:     80,      960,      2016,      1024;
  4:    356,     5696,     18288,     19456,       6400;
  5:   1584,    31680,    137088,    224256,     153600,      36864;
  6:   7048,   169152,    921312,   2040832,    2150400,    1069056,     200704;
  7:  31360,   878080,   5766912,  16138240,   23065600,   17620992,    6823936,   1048576;
  8: 139536,  4465152,  34335360, 116195328,  210182400,  217866240,  129454080,  40894464,   5308416;
  9: 620864, 22351104, 196964352, 782229504, 1713945600, 2250215424, 1813561344, 877658112, 233570304, 26214400;
%F Formula
The row polynomials satisfy (3+(18*n-3)*x+(33*n^2-3*n-7)*x^2+(18*n^3+15*n^2-27*n)*x^3+(3*n^4+6*n^3-17*n^2-8*n+4)*x^4-(4*n^3+4*n^2)*x^5)*P(n,x) = (6+(36*n+42)*x+(66*n^2+198*n+106)*x^2+(36*n^3+270*n^2+366*n+104)*x^3+(6*n^4+108*n^3+314*n^2+204*n)*x^4+(12*n^4+64*n^3+44*n^2-88*n-64)*x^5-(16*n^3+64*n^2+80*n+32)*x^6)*P(n-2,x) + (12+(72*n+36)*x+(132*n^2+192*n+8)*x^2+(72*n^3+300*n^2+120*n-100)*x^3+(12*n^4+120*n^3+160*n^2-112*n-132)*x^4+(12*n^4+32*n^3-60*n^2-144*n-64)*x^5-(16*n^3+48*n^2+48*n+16)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((6 + (36*n + 42)*x + (66*n^2 + 198*n + 106)*x^2 + (36*n^3 + 270*n^2 + 366*n + 104)*x^3 + (6*n^4 + 108*n^3 + 314*n^2 + 204*n)*x^4 + (12*n^4 + 64*n^3 + 44*n^2 - 88*n - 64)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*P[n - 2, x] + (12 + (72*n + 36)*x + (132*n^2 + 192*n + 8)*x^2 + (72*n^3 + 300*n^2 + 120*n - 100)*x^3 + (12*n^4 + 120*n^3 + 160*n^2 - 112*n - 132)*x^4 + (12*n^4 + 32*n^3 - 60*n^2 - 144*n - 64)*x^5 - (16*n^3 + 48*n^2 + 48*n + 16)*x^6)*P[n - 1, x])/(3 + (18*n - 3)*x + (33*n^2 - 3*n - 7)*x^2 + (18*n^3 + 15*n^2 - 27*n)*x^3 + (3*n^4 + 6*n^3 - 17*n^2 - 8*n + 4)*x^4 - (4*n^3 + 4*n^2)*x^5)]];
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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((6 + (36*n + 42)*x + (66*n^2 + 198*n + 106)*x^2 + (36*n^3 + 270*n^2 + 366*n + 104)*x^3 + (6*n^4 + 108*n^3 + 314*n^2 + 204*n)*x^4 + (12*n^4 + 64*n^3 + 44*n^2 - 88*n - 64)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*v[i - 2] + (12 + (72*n + 36)*x + (132*n^2 + 192*n + 8)*x^2 + (72*n^3 + 300*n^2 + 120*n - 100)*x^3 + (12*n^4 + 120*n^3 + 160*n^2 - 112*n - 132)*x^4 + (12*n^4 + 32*n^3 - 60*n^2 - 144*n - 64)*x^5 - (16*n^3 + 48*n^2 + 48*n + 16)*x^6)*v[i - 1])/(3 + (18*n - 3)*x + (33*n^2 - 3*n - 7)*x^2 + (18*n^3 + 15*n^2 - 27*n)*x^3 + (3*n^4 + 6*n^3 - 17*n^2 - 8*n + 4)*x^4 - (4*n^3 + 4*n^2)*x^5)); 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.