Real-rooted atlas

A038329 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 4\,x + 2\,n x\right)\,P_{n}(x) = \left(75 + 120\,x + 36\,x^{2} + 30\,n x + 12\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 360\,x + 108\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0225360108693
133758100486086417199
250625162000145800518406480416745
3759375303750036450001944000486000466569918531
411390625546750008201250058320000218700004199040326592232793757
5170859375956812500172226250015309000007654500002204496003429216022394885403265623
62562890625164025000003444525000036741600000229635000008817984000205752960026873856015116544124275109329
738443359375276792187500664301250000826686000000620014500000297606960000925888320001813985280020407334401007769602836714452075

All 15 rows

Technical details
Node id
n-d8e5c4e43f9340d6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65a6331fbe5d08da

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:           225,            360,             108;
  1:          3375,           8100,            4860,             864;
  2:         50625,         162000,          145800,           51840,            6480;
  3:        759375,        3037500,         3645000,         1944000,          486000,           46656;
  4:      11390625,       54675000,        82012500,        58320000,        21870000,         4199040,          326592;
  5:     170859375,      956812500,      1722262500,      1530900000,       765450000,       220449600,        34292160,        2239488;
  6:    2562890625,    16402500000,     34445250000,     36741600000,     22963500000,      8817984000,      2057529600,      268738560,      15116544;
  7:   38443359375,   276792187500,    664301250000,    826686000000,    620014500000,    297606960000,     92588832000,    18139852800,    2040733440,     100776960;
  8:  576650390625,  4613203125000,  12455648437500,  17714700000000,  15500362500000,   8928208800000,   3472081200000,   906992640000,  153055008000,   15116544000,    665127936;
  9: 8649755859375, 76117851562500, 228353554687500, 365365687500000, 365365687500000, 245525742000000, 114578679600000, 37413446400000, 8418025440000, 1247114880000, 109746109440, 4353564672;
%F Formula
The row polynomials satisfy (5+(2*n+4)*x)*P(n,x) = (75+(30*n+120)*x+(12*n+36)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 360*x + 108*x^2.
%t Mathematica
P[0, x_] := 225 + 360*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((75 + (30*n + 120)*x + (12*n + 36)*x^2)*P[n - 1, x])/(5 + (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 225 + 360*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (30*n + 120)*x + (12*n + 36)*x^2)*v[i - 1])/(5 + (2*n + 4)*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.