Real-rooted atlas

A085880 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 8\,n x + n x^{2} + 6\,n^{2} x - 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-48 + 12\,x^{2} + 32\,n - 72\,n x - 44\,n x^{2} + 48\,n^{2} x + 12\,n^{2} x^{2} - 6\,n^{2} x^{3} + 8\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
38321858
44024027080630
52241792302417923507182
6134413440302402688010500151283916
78448101376285120337920198000570246468994356
85491276876825945923843840300300012972962942942745611884158
936608058572802306304041000960400400002306304078478401464320115830142818390
102489344448081922016368644182097924900896003528645121600959364480819270887964862001722577428
111719910434398208017414092804127784960564345600048759459842765293440103194624024488568033592000203231620827527084
121203937282648662016148987238403972993024062078016000625746401284258551897619864965120628539912012932920001564883328465184252244494684
1385201715220448411648126524547072374887546880658982016000759147282432602748883968337398792192133443858240366101120006644735328718889472351540283058442246412
146085836800158231756800106806435840034810986496006799020800000881153095680079956484608005221647974400247824308160084987760000020567037920033377011200326430260014560840037111906776600
1543818024960122690469888089717406105603189952217088068534129664000986891467161601007451706060807519173083136041634483770880171335324160005182893555840112146757632016452085104014677326720601749000450554340568680
Technical details
Node id
n-44700717a696a0ec · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        4;
   3:       8,       32,        18;
   4:      40,      240,       270,        80;
   5:     224,     1792,      3024,      1792,       350;
   6:    1344,    13440,     30240,     26880,     10500,      1512;
   7:    8448,   101376,    285120,    337920,    198000,     57024,      6468;
   8:   54912,   768768,   2594592,   3843840,   3003000,   1297296,    294294,    27456;
   9:  366080,  5857280,  23063040,  41000960,  40040000,  23063040,   7847840,  1464320,  115830;
  10: 2489344, 44808192, 201636864, 418209792, 490089600, 352864512, 160095936, 44808192, 7088796, 486200;
%F Formula
The row polynomials satisfy (4*n+(6*n^2-8*n)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (32*n-48+(48*n^2-72*n)*x+(8*n^3+12*n^2-44*n+12)*x^2+(4*n^3-6*n^2)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n - 48 + (48*n^2 - 72*n)*x + (8*n^3 + 12*n^2 - 44*n + 12)*x^2 + (4*n^3 - 6*n^2)*x^3)*P[n - 1, x])/(4*n + (6*n^2 - 8*n)*x + (n^3 - 2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n - 48 + (48*n^2 - 72*n)*x + (8*n^3 + 12*n^2 - 44*n + 12)*x^2 + (4*n^3 - 6*n^2)*x^3)*v[i - 1])/(4*n + (6*n^2 - 8*n)*x + (n^3 - 2*n^2 + n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.