Real-rooted atlas

A038305 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 + 390\,x + 144\,x^{2} + 234\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 9126\,x + 7956\,x^{2} + 1944\,x^{3} + 3042\,n x + 4212\,n x^{2} + 1296\,n x^{3} + 468\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 624\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01696243241117
121971216812636345630457
22856121091232853617971232400780121
3371293342732071182805840640210600027993619143469
448268095346619213880646015185664082134000218350082286144455211253
5627485178109039122526277572345473856024913980009934928642080391041791590410565514433
68157307211204771526443788811248718585620486477634800034441085952108180334081863254016136048896240545589553
7106044993731761978357367318987022881401241959936151576654320010074017640964219033029121090003599361591772083210077696005390940457909

All 15 rows

Technical details
Node id
n-dc4135fd9023e99e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-53bee5cda939f818

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:           169,            624,             324;
  1:          2197,          12168,           12636,            3456;
  2:         28561,         210912,          328536,          179712,           32400;
  3:        371293,        3427320,         7118280,         5840640,         2106000,          279936;
  4:       4826809,       53466192,       138806460,       151856640,        82134000,        21835008,         2286144;
  5:      62748517,      810903912,      2526277572,      3454738560,      2491398000,       993492864,       208039104,        17915904;
  6:     815730721,    12047715264,     43788811248,     71858562048,     64776348000,     34441085952,     10818033408,      1863254016,      136048896;
  7:   10604499373,   176197835736,    731898702288,   1401241959936,   1515766543200,   1007401764096,    421903302912,    109000359936,    15917720832,    1007769600;
  8:  137858491849,  2545079849520,  11893353912180,  26023064970240,  32841608436000,  26192445866496,  13711857344640,   4723348930560,  1034651854080,  131010048000,    7316407296;
  9: 1792160394037, 36394641848136, 188972178826860, 465162286343040, 670907143764000, 624253293151488, 392159120056704, 168859724267520, 49318405044480, 9367218432000, 1046246243328, 52242776064;
%F Formula
The row polynomials satisfy (169+(234*n+390)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (2197+(3042*n+9126)*x+(468*n^2+4212*n+7956)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 624*x + 324*x^2.
%t Mathematica
P[0, x_] := 169 + 624*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 169 + 624*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*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

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.