Real-rooted atlas

A038226 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 440\,x + 256\,x^{2} + 264\,n x + 256\,n x^{2} + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 8712\,x + 11968\,x^{2} + 4608\,x^{3} + 2904\,n x + 6336\,n x^{2} + 3072\,n x^{3} + 704\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 704\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01217045761401
113311161619008819240147
2146411703684181763604481024001066033
31610512342560766656099123205632000117964826894139
41771561309217921264982402180710401858560007785676812845056653820457
519487171396829664194807289641978675204770304000299748556898906931213421772815453333859
6214358881498871577628571735808738824683521049466880008792624332843519049728118111600641358954496357219374433
72357947691617353577284040859778561219060727808207794442240021761745223681436128641024584652423168134536495104134217728008110098287947

All 15 rows

Technical details
Node id
n-f95ad06430ea3396 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-46efc6967b68e78c

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,           704,            576;
  1:         1331,         11616,          19008,            8192;
  2:        14641,        170368,         418176,          360448,          102400;
  3:       161051,       2342560,        7666560,         9912320,         5632000,         1179648;
  4:      1771561,      30921792,      126498240,       218071040,       185856000,        77856768,        12845056;
  5:     19487171,     396829664,     1948072896,      4197867520,      4770304000,      2997485568,       989069312,       134217728;
  6:    214358881,    4988715776,    28571735808,     73882468352,    104946688000,     87926243328,     43519049728,     11811160064,      1358954496;
  7:   2357947691,   61735357728,   404085977856,   1219060727808,   2077944422400,   2176174522368,   1436128641024,    584652423168,    134536495104,    13421772800;
  8:  25937424601,  754543261120,  5556182195520,  19156668579840,  38095647744000,  47875839492096,  39493537628160,  21437255516160,   7399507230720,  1476395008000,   129922760704;
  9: 285311670611, 9129973459552, 74699782850880, 289744612270080, 658510482432000, 965496096423936, 955743610601472, 648476979363840, 298446791639040, 89321897984000, 15720654045184, 1236950581248;
%F Formula
The row polynomials satisfy (121+(264*n+440)*x+(64*n^2+256*n+256)*x^2)*P(n,x) = (1331+(2904*n+8712)*x+(704*n^2+6336*n+11968)*x^2+(512*n^2+3072*n+4608)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 704*x + 576*x^2.
%t Mathematica
P[0, x_] := 121 + 704*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (2904*n + 8712)*x + (704*n^2 + 6336*n + 11968)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*P[n - 1, x])/(121 + (264*n + 440)*x + (64*n^2 + 256*n + 256)*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 + 704*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (2904*n + 8712)*x + (704*n^2 + 6336*n + 11968)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*v[i - 1])/(121 + (264*n + 440)*x + (64*n^2 + 256*n + 256)*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.