Real-rooted atlas

A038319 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(121 + 220\,x + 75\,x^{2} + 55\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 220\,x + 75\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012122075416
11331363024755007936
2146415324054450220003125147456
3161051732050998250605000171875187502686976
41771561966306016471125133100005671875123750010937548234496
519487171124009270253655325256217500145578125476437508421875625000855638016
62143588811558973680372027810045094280003202718750139755000037056250055000000351562515032385536
72357947691192922992905261536170074405562000634138312503458936250012228562500272250000034804687519531250261993005056

All 15 rows

Technical details
Node id
n-4b78239f1284a1b5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0df1d6c1b80d1c27

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          121,           220,            75;
  1:         1331,          3630,          2475,            500;
  2:        14641,         53240,         54450,          22000,           3125;
  3:       161051,        732050,        998250,         605000,         171875,          18750;
  4:      1771561,       9663060,      16471125,       13310000,        5671875,        1237500,        109375;
  5:     19487171,     124009270,     253655325,      256217500,      145578125,       47643750,       8421875,        625000;
  6:    214358881,    1558973680,    3720278100,     4509428000,     3202718750,     1397550000,     370562500,      55000000,      3515625;
  7:   2357947691,   19292299290,   52615361700,    74405562000,    63413831250,    34589362500,   12228562500,    2722500000,    348046875,     19531250;
  8:  25937424601,  235794769100,  723461223375,  1169230260000,  1162586906250,   760965975000,  336285468750,   99825000000,  19142578125,   2148437500,   107421875;
  9: 285311670611, 2853116706110, 9726534225375, 17684607682500, 20096145093750, 15346147162500, 8138108343750, 3019706250000, 772083984375, 129980468750, 12998046875, 585937500;
%F Formula
The row polynomials satisfy (11+(5*n+10)*x)*P(n,x) = (121+(55*n+220)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 220*x + 75*x^2.
%t Mathematica
P[0, x_] := 121 + 220*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (55*n + 220)*x + (25*n + 75)*x^2)*P[n - 1, x])/(11 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 220*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (55*n + 220)*x + (25*n + 75)*x^2)*v[i - 1])/(11 + (5*n + 10)*x)); 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.