Real-rooted atlas

A038319 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 275\,x + 100\,x^{2} + 165\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 5445\,x + 4675\,x^{2} + 1125\,x^{3} + 1815\,n x + 2475\,n x^{2} + 750\,n x^{3} + 275\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 440\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0121440225786
1133172607425200018016
2146411064801633508800015625388096
31610511464100299475024200008593751125008011776
41771561193261204941337553240000283593757425000765625160301056
51948717124801854076096597510248700007278906252858625005895312550000003131047936
62143588813117947360111608343001803771200016013593750838530000025939375004400000003164062559995324416
7235794769138584598580157846085100297622248000317069156250207536175000855999375002178000000031324218751953125001131723882496

All 15 rows

Technical details
Node id
n-dcf2d7cd5a114696 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-44310a99e9adba32

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          121,           440,            225;
  1:         1331,          7260,           7425,           2000;
  2:        14641,        106480,         163350,          88000,           15625;
  3:       161051,       1464100,        2994750,        2420000,          859375,         112500;
  4:      1771561,      19326120,       49413375,       53240000,        28359375,        7425000,         765625;
  5:     19487171,     248018540,      760965975,     1024870000,       727890625,      285862500,       58953125,        5000000;
  6:    214358881,    3117947360,    11160834300,    18037712000,     16013593750,     8385300000,     2593937500,      440000000,      31640625;
  7:   2357947691,   38584598580,   157846085100,   297622248000,    317069156250,   207536175000,    85599937500,    21780000000,    3132421875,     195312500;
  8:  25937424601,  471589538200,  2170383670125,  4676921040000,   5812934531250,  4565795850000,  2353998281250,   798600000000,  172283203125,   21484375000,   1181640625;
  9: 285311670611, 5706233412220, 29179602676125, 70738430730000, 100480725468750, 92076882975000, 56966758406250, 24157650000000, 6948755859375, 1299804687500, 142978515625, 7031250000;
%F Formula
The row polynomials satisfy (121+(165*n+275)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1331+(1815*n+5445)*x+(275*n^2+2475*n+4675)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 440*x + 225*x^2.
%t Mathematica
P[0, x_] := 121 + 440*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (1815*n + 5445)*x + (275*n^2 + 2475*n + 4675)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(121 + (165*n + 275)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 440*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (1815*n + 5445)*x + (275*n^2 + 2475*n + 4675)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(121 + (165*n + 275)*x + (25*n^2 + 100*n + 100)*x^2)); 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

Same fingerprint ?

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.