Real-rooted atlas

A094638 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(81\,n x^{2} + 12\,n^{2} x - 162\,n^{2} x^{2} - n^{3} - 12\,n^{3} x + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(36\,n x^{3} - 12\,n^{2} x^{2} - 72\,n^{2} x^{3} + 12\,n^{3} x^{2} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 12\,n x + 2\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121214
2684162252
324552234925925517
41203912266496307248600142353
5720306722919241089072176985010497604231998
65040266544328568416965072430191005322283225719120142483392
740320255571238879676258386112900819225172668024017270389087054387205359838913
83628802687212848792553239843411841779685222546672434576719482380126048633139221427701120222831059049
93628800307912320651173709663263837376346057915950115866596026824346943507883142139952384228120836673671425670400010147110365718
10399168003822491520924131896561043194856256677212945245027695027531988740938312362781299638079152641443964183511769234602251200025927518355200502334225808588
114790016005113891584013927782712321793991206822413505528820510065415192196428021330942948137974753161033167232716077668517506069909693140808003998506899166560101824435722240026863344102052125
1262270208007337821363202224836970905632235666624576027648388005039001550524112608802459783670008829081162090057824051328309655996480957788409139342646386400356507213010524928184545108588349440430208240926464001543357637623449225
13871782912001124436515328037586655838886460555469606018565835135255640560037234672280153136016600412632334464085314567755292583616123358478986424425502063731359057700680024298386561630323208191302803488906772489049074944976293760194586496665200640094808518150315152150
14130767436800018331143022080067005908828179201188979074353963521272665557635580800911429064463438108846186439935322572200170783637055337814336467540210721908036088949196261566041012000141326407509757833369615014719070907171402241078699526372047372416469968950517796961280938184894635788800006201442237140020887200
Technical details
Node id
n-212d41ac22bbd28e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7b2795e7b686b0fb

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,        12;
  2:       6,        84,        162;
  3:      24,       552,       2349,        2592;
  4:     120,      3912,      26649,       63072,        48600;
  5:     720,     30672,     291924,     1089072,      1769850,       1049760;
  6:    5040,    266544,    3285684,    16965072,     43019100,      53222832,      25719120;
  7:   40320,   2555712,   38879676,   258386112,    900819225,    1726680240,    1727038908,     705438720;
  8:  362880,  26872128,  487925532,  3984341184,  17796852225,   46672434576,   71948238012,   60486331392,   21427701120;
  9: 3628800, 307912320, 6511737096, 63263837376, 346057915950, 1158665960268, 2434694350788, 3142139952384, 2281208366736, 714256704000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(12*n^3-12*n^2)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((12*n^3-12*n^2)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (12*n^3 - 12*n^2)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((12*n^3 - 12*n^2)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (12*n^3 - 12*n^2)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((12*n^3 - 12*n^2)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.