Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 12\,x + 9\,x^{2} + 18\,n x - 36\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 12\,x + 54\,x^{2} + 108\,x^{3} + 81\,x^{4} + 18\,n x + 144\,n x^{2} + 378\,n x^{3} + 324\,n x^{4} + 36\,n^{2} x^{2} + 216\,n^{2} x^{3} + 324\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112481106
2148486172820254288
317212158640303755248835721128512
4196226824192141750489888100018811197445314413309568
5112036455184042525022044967501410167961602391484519683000714492977725696
611445346950401002375692841633006204110854656263063295433026000471565314306110016898135291715470336
711687371157248202702517513496107270163480370176159591732339405366007152073929928533715281730311394374822312107616802536205232128

All 15 rows

Technical details
Node id
n-f5469f7bf3514e04 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1b92fdc3be2f687b

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  24,    81;
  2: 1,  48,   486,   1728,    2025;
  3: 1,  72,  1215,   8640,   30375,    52488,     35721;
  4: 1,  96,  2268,  24192,  141750,   489888,   1000188,    1119744,      531441;
  5: 1, 120,  3645,  51840,  425250,  2204496,   7501410,   16796160,    23914845,    19683000,      7144929;
  6: 1, 144,  5346,  95040, 1002375,  6928416,  33006204,  110854656,   263063295,   433026000,    471565314,    306110016,      89813529;
  7: 1, 168,  7371, 157248, 2027025, 17513496, 107270163,  480370176,  1595917323,  3940536600,   7152073929,   9285337152,    8173031139,    4374822312,    1076168025;
  8: 1, 192,  9720, 241920, 3685500, 38211264, 286053768, 1601233920,  6839645670, 22517352000,  57216591432, 111424045824,  163460622780,  174992892480,  129140163000,   58773123072,   12440502369;
  9: 1, 216, 12393, 352512, 6196500, 74952864, 663124644, 4454341632, 23254795278, 95698746000, 312647803182, 811803762432, 1667298352356, 2677391254944, 3293074156500, 2997429276672, 1903396862457, 753145430616, 139858796529;
%F Formula
The row polynomials satisfy (1+(18*n-12)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (1+(18*n+12)*x+(36*n^2+144*n+54)*x^2+(216*n^2+378*n+108)*x^3+(324*n^2+324*n+81)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*P[n - 1, x])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*v[i - 1])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.